slide 1 / 210content.njctl.org/courses/math/archived-courses... · slide 8 / 210 1 the letter on...
TRANSCRIPT
![Page 2: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/2.jpg)
Slide 2 / 210
Table of Contents
· Questions from Released PARCC Test
· Triangle Sum Theorem
· Exterior Angle Theorem
· Triangles
· Inequalities in Triangles
· Similar Triangles
Click on the topic to go to that section
![Page 3: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/3.jpg)
Slide 3 / 210
www.njctl.org
Geometry
Triangles
![Page 5: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/5.jpg)
Slide 5 / 210
Geometric Figures
Euclid now makes the transitions to geometric figures, which are created by a boundary which separates space into that which is within the figure and that which is not.
Definition 13. A boundary is that which is an extremity of anything.
Definition 14. A figure is that which is contained by any boundary or boundaries.
![Page 6: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/6.jpg)
Slide 6 / 210
Geometric Figures
His definitions from 15 to 18 relate to circles, which we will discuss later. In this chapter, we will be discussing triangles, which are an example of a rectilinear figure: a figure bounded by straight lines.
A triangle is bounded by three lines.
Definition 19. Rectilinear figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.
![Page 7: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/7.jpg)
Slide 7 / 210
Parts of a TriangleEach triangle has three sides and three vertices.
Each vertex is where two sides meet.
A pair of sides and the vertex define an angle, so each triangle includes three angles.
Write "side" next to each side and circle the vertices on the triangle below.
![Page 8: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/8.jpg)
Slide 8 / 210
1 The letter on this triangle that corresponds to a side is:
A
B
C
![Page 9: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/9.jpg)
Slide 8 (Answer) / 210
1 The letter on this triangle that corresponds to a side is:
A
B
C
[This object is a pull tab]
Ans
wer
C
![Page 10: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/10.jpg)
Slide 9 / 210
2 The letter on this triangle that represents a vertex is:
AB
C
![Page 11: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/11.jpg)
Slide 9 (Answer) / 210
2 The letter on this triangle that represents a vertex is:
AB
C
[This object is a pull tab]
Ans
wer
A
![Page 12: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/12.jpg)
Slide 10 / 210
Parts of a Triangle
C
A B
Each vertex is named with a letter.
The sides can then be named with the letters of the two vertices on either side of it.
The triangle is named with a triangle symbol Δ in front followed by the three letters of its vertices.
Name the 3 sides of this triangle
______ ______ ______
![Page 13: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/13.jpg)
Slide 11 / 210
3 What is the name of the side shown in red?
A ABB BCC AC
C
A B
![Page 14: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/14.jpg)
Slide 11 (Answer) / 210
3 What is the name of the side shown in red?
A ABB BCC AC
C
A B[This object is a pull tab]
Ans
wer
C
![Page 15: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/15.jpg)
Slide 12 / 210
4 What is the name of the side shown in red?
A AB
B BC
C AC
C
A B
![Page 16: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/16.jpg)
Slide 12 (Answer) / 210
4 What is the name of the side shown in red?
A AB
B BC
C AC
C
A B
[This object is a pull tab]
Ans
wer
A
![Page 17: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/17.jpg)
Slide 13 / 210
5 Which of the following are names of this triangle?
A ΔABCB ΔBCAC ΔACB
C
A B
D ΔCABE all of these
![Page 18: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/18.jpg)
Slide 13 (Answer) / 210
5 Which of the following are names of this triangle?
A ΔABCB ΔBCAC ΔACB
C
A B
D ΔCABE all of these
[This object is a pull tab]
Ans
wer
E
![Page 19: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/19.jpg)
Slide 14 / 210
Parts of a Triangle
C
A B
In the above, the red side is ________________ A,
while the green sides are ________________ to A.
A side is opposite an angle if it does not touch it. Otherwise, it is adjacent to the angle.
![Page 20: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/20.jpg)
Slide 15 / 210
6 Which side is opposite angle B?
A AB
B CA
C BC
D None
C
A B
![Page 21: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/21.jpg)
Slide 15 (Answer) / 210
6 Which side is opposite angle B?
A AB
B CA
C BC
D None
C
A B[This object is a pull tab]
Ans
wer
B
![Page 22: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/22.jpg)
Slide 16 / 210
7 Which side is opposite angle A?
A AB B CA
C BC D None
C
A B
![Page 23: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/23.jpg)
Slide 16 (Answer) / 210
7 Which side is opposite angle A?
A AB B CA
C BC D None
C
A B[This object is a pull tab]
Ans
wer
C
![Page 24: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/24.jpg)
Slide 17 / 210
8 Which sides are adjacent to angle C?
A AB & BC
B CA & BA
C BC & CAD None
C
A B
![Page 25: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/25.jpg)
Slide 17 (Answer) / 210
8 Which sides are adjacent to angle C?
A AB & BC
B CA & BA
C BC & CAD None
C
A B
[This object is a pull tab]
Ans
wer
C
![Page 26: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/26.jpg)
Slide 18 / 210
9 Which sides are adjacent to angle B?
A AB & BCB CA & BA C BC & CAD None
C
A B
![Page 27: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/27.jpg)
Slide 18 (Answer) / 210
9 Which sides are adjacent to angle B?
A AB & BCB CA & BA C BC & CAD None
C
A B[This object is a pull tab]
Ans
wer
A
![Page 28: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/28.jpg)
Slide 19 / 210
Types of Triangles
In general, a triangle can have sides of all different lengths and angles of all different measure.
However, there are names given to triangles which have specific or special angles or some number of equal sides or angles.
Euclid defined the names for a number of these in his definitions.
![Page 29: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/29.jpg)
Slide 20 / 210
Definition 20: Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle is that which has two of its sides alone equal, and a scalene triangle is that which has its three sides unequal
Classifying Triangles
Triangles can be classified by their sides or by their angles.
In this definition, Euclid used the sides.
In his next definition, Euclid uses the angles.
![Page 30: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/30.jpg)
Slide 21 / 210
Definition 21: Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle is that which has an obtuse angle, and an acute-angled triangle is that which has
its three angles acute.
Classifying Triangles
We will draw from both definitions, since in several cases both definitions apply to the same triangle.
![Page 31: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/31.jpg)
Slide 22 / 210
Acute Triangles
In an acute triangle, every angle of a triangle is acute.
Notice that no angle is equal to or greater than 90º in this triangle.
Classifying Triangles
Definition 21: "...an acute-angled triangle is that which has its three angles acute."
![Page 32: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/32.jpg)
Slide 23 / 210
Right Triangles
A right triangle has one right angle and two acute angles.
Notice that one angle is 90º, which means that the other two sum to 90º; and they are acute.
The side opposite the right angle is called the hypotenuse and the other two sides are called the legs.
Classifying Triangles
Definition 21: "...a right-angled triangle is that which has a right angle..."
![Page 33: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/33.jpg)
Slide 24 / 210
Isosceles Triangles
An isosceles triangle has two sides with equal length.
The angles opposite those equal sides are of equal measure.
x x
Classifying Triangles
Definition 20: "...an isosceles triangle is that which has two of its sides alone equal..."
![Page 34: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/34.jpg)
Slide 25 / 210
Isosceles Triangles
The equal angles, of measure x in this diagram, are called the base angles. The side between them is called the base.
The other two sides, opposite the base angles and congruent to each other are called the legs.
This is a special case of an acute triangle.
x x
Classifying Triangles
![Page 35: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/35.jpg)
Slide 26 / 210
Obtuse Triangles
An obtuse triangle has one angle which is greater than 90º and two acute angles.
Notice that one angle is greater than 90º, which means that the other two sum to less than 90º; and they are acute..
Classifying Triangles
Definition 21: "...an obtuse-angled triangle is that which has an obtuse angle..."
![Page 36: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/36.jpg)
Slide 27 / 210
Equiangular / Equilateral Triangles
An equiangular, or equilateral, triangle has angles of equal measure and sides of equal length.
Definition 20: "...an equilateral triangle is that which has its three sides equal..."
All the angles are of equal measure and all the sides are of equal length.
Each angle measures 60º.
This is a special acute triangle.x x
x
Classifying Triangles
![Page 37: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/37.jpg)
Slide 28 / 210
Classifying Triangles
Scalene Triangles
None of the sides or angles of a scalene triangle are congruent with one another.
Definition 20: "...a scalene triangle is that which has its three sides unequal..."
Note that in this triangle none of the sides or angles are equal.
![Page 38: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/38.jpg)
Slide 29 / 210
10 An isosceles triangle is _______________ an equilateral triangle.
A SometimesB AlwaysC Never
![Page 39: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/39.jpg)
Slide 29 (Answer) / 210
10 An isosceles triangle is _______________ an equilateral triangle.
A SometimesB AlwaysC Never
[This object is a pull tab]
Ans
wer
C
![Page 40: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/40.jpg)
Slide 30 / 210
11 An obtuse triangle is _______________ an isosceles triangle.
A SometimesB AlwaysC Never
![Page 41: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/41.jpg)
Slide 30 (Answer) / 210
11 An obtuse triangle is _______________ an isosceles triangle.
A SometimesB AlwaysC Never
[This object is a pull tab]
Ans
wer
A
![Page 42: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/42.jpg)
Slide 31 / 210
12 A triangle can have more than one obtuse angle.
TrueFalse
![Page 43: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/43.jpg)
Slide 31 (Answer) / 210
12 A triangle can have more than one obtuse angle.
TrueFalse
[This object is a pull tab]
Ans
wer
False
![Page 44: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/44.jpg)
Slide 32 / 210
13 A triangle can have more than one right angle.
TrueFalse
![Page 45: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/45.jpg)
Slide 32 (Answer) / 210
13 A triangle can have more than one right angle.
TrueFalse
[This object is a pull tab]
Ans
wer
False
![Page 46: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/46.jpg)
Slide 33 / 210
14 Each angle in an equiangular triangle measures 60°
TrueFalse
![Page 47: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/47.jpg)
Slide 33 (Answer) / 210
14 Each angle in an equiangular triangle measures 60°
TrueFalse
[This object is a pull tab]
Ans
wer
True
![Page 48: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/48.jpg)
Slide 34 / 210
15 An equilateral triangle is also an isosceles triangle
TrueFalse
![Page 49: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/49.jpg)
Slide 34 (Answer) / 210
15 An equilateral triangle is also an isosceles triangle
TrueFalse
[This object is a pull tab]
Ans
wer False
![Page 50: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/50.jpg)
Slide 35 / 210
16 This triangle is classified as _____. (Choose all that apply.)
A acute
B right
C isosceles
D obtuse
E equilateral
F equiangular
G scalene
60º8.6
60º
60º8.68.6
![Page 51: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/51.jpg)
Slide 35 (Answer) / 210
16 This triangle is classified as _____. (Choose all that apply.)
A acute
B right
C isosceles
D obtuse
E equilateral
F equiangular
G scalene
60º8.6
60º
60º8.68.6
[This object is a pull tab]
Ans
wer
A, E, F
![Page 52: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/52.jpg)
Slide 36 / 210
17 This triangle is classified as _____. (Choose all that apply.)
A acute
B right
C isosceles
D obtuse
E equilateral
F equiangular
G scalene
57º
79º 44º
6.1 8.7
7.4
![Page 53: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/53.jpg)
Slide 36 (Answer) / 210
17 This triangle is classified as _____. (Choose all that apply.)
A acute
B right
C isosceles
D obtuse
E equilateral
F equiangular
G scalene
57º
79º 44º
6.1 8.7
7.4[This object is a pull tab]
Ans
wer
A, G
![Page 54: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/54.jpg)
Slide 37 / 210
18 This triangle is classified as _____. (Choose all that apply.)
A acute
B right
C isosceles
D obtuse
E equilateral
F equiangular
G scalene
26°
128° 26°
2.5
2.5
4.5
![Page 55: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/55.jpg)
Slide 37 (Answer) / 210
18 This triangle is classified as _____. (Choose all that apply.)
A acute
B right
C isosceles
D obtuse
E equilateral
F equiangular
G scalene
26°
128° 26°
2.5
2.5
4.5
[This object is a pull tab]
Ans
wer
C, D
![Page 56: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/56.jpg)
Slide 38 / 210
19 This triangle is classified as _____. Choose all that apply.
A acute
B right
C isosceles
D obtuse
E equilateral
F equiangular
G scalene
4.8 4.8
45° 45°
6.8
![Page 57: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/57.jpg)
Slide 38 (Answer) / 210
19 This triangle is classified as _____. Choose all that apply.
A acute
B right
C isosceles
D obtuse
E equilateral
F equiangular
G scalene
4.8 4.8
45° 45°
6.8[This object is a pull tab]
Ans
wer
B, C
![Page 58: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/58.jpg)
Slide 39 / 210
Measure and Classify the triangle by sides and anglesExample
isosceles, acuteClick for AnswerClick for Answer
![Page 59: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/59.jpg)
Slide 40 / 210
Measure and Classify the triangle by sides and anglesExample
scalene, obtuseClick for AnswerClick for Answer
![Page 60: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/60.jpg)
Slide 41 / 210
Measure and Classify the triangle by sides and anglesExample
scalene, acuteClick for AnswerClick for Answer
![Page 61: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/61.jpg)
Slide 42 / 210
20 Classify the triangle with the given information: Side lengths: 3 cm, 4 cm, 5 cm
A EquilateralB IsoscelesC Scalene
D AcuteE EquiangularF RightG Obtuse
![Page 62: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/62.jpg)
Slide 42 (Answer) / 210
20 Classify the triangle with the given information: Side lengths: 3 cm, 4 cm, 5 cm
A EquilateralB IsoscelesC Scalene
D AcuteE EquiangularF RightG Obtuse
[This object is a pull tab]
Ans
wer
C Scalene
Bonus: F Right32+42 = 52
9+16 = 25
![Page 63: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/63.jpg)
Slide 43 / 210
21 Classify the triangle with the given information: Side lengths: 3 cm, 2 cm, 3 cm
A EquilateralB IsoscelesC Scalene
D AcuteE EquiangularF RightG Obtuse
![Page 64: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/64.jpg)
Slide 43 (Answer) / 210
21 Classify the triangle with the given information: Side lengths: 3 cm, 2 cm, 3 cm
A EquilateralB IsoscelesC Scalene
D AcuteE EquiangularF RightG Obtuse
[This object is a pull tab]
Ans
wer
B Isosceles
Bonus: D Acute
3 cm 3 cm
2 cm
![Page 65: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/65.jpg)
Slide 44 / 210
22 Classify the triangle with the given information: Side lengths: 5 cm, 5 cm, 5 cm
A EquilateralB IsoscelesC Scalene
D AcuteE EquiangularF RightG Obtuse
![Page 66: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/66.jpg)
Slide 44 (Answer) / 210
22 Classify the triangle with the given information: Side lengths: 5 cm, 5 cm, 5 cm
A EquilateralB IsoscelesC Scalene
D AcuteE EquiangularF RightG Obtuse
[This object is a pull tab]
Ans
wer
A Equilateral
Bonus:E Equiangular (all equilateral triangles are equiangular)
D Acute (all angles are 60o )
![Page 67: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/67.jpg)
Slide 45 / 210
23 Classify the triangle with the given information: Angle Measures: 25°, 120°, 35°
A EquilateralB IsoscelesC Scalene
D AcuteE EquiangularF RightG Obtuse
![Page 68: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/68.jpg)
Slide 45 (Answer) / 210
23 Classify the triangle with the given information: Angle Measures: 25°, 120°, 35°
A EquilateralB IsoscelesC Scalene
D AcuteE EquiangularF RightG Obtuse
[This object is a pull tab]
Ans
wer
G Obtuse
Bonus: C Scalene (all angles are different, so all sides are different)
![Page 69: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/69.jpg)
Slide 46 / 210
24 Classify the triangle with the given information: Angle Measures: 30°, 60°, 90°
A EquilateralB IsoscelesC Scalene
D AcuteE EquiangularF RightG Obtuse
![Page 70: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/70.jpg)
Slide 46 (Answer) / 210
24 Classify the triangle with the given information: Angle Measures: 30°, 60°, 90°
A EquilateralB IsoscelesC Scalene
D AcuteE EquiangularF RightG Obtuse
[This object is a pull tab]
Ans
wer
F Right
Bonus: C Scalene(all angles are different, so all sides are different)
![Page 71: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/71.jpg)
Slide 47 / 210
25 Classify the triangle with the given information: Side lengths: 3 cm, 4 cm, 5 cmAngle measures: 37°, 53°, 90°
A EquilateralB IsoscelesC Scalene
D AcuteE EquiangularF RightG Obtuse
![Page 72: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/72.jpg)
Slide 47 (Answer) / 210
25 Classify the triangle with the given information: Side lengths: 3 cm, 4 cm, 5 cmAngle measures: 37°, 53°, 90°
A EquilateralB IsoscelesC Scalene
D AcuteE EquiangularF RightG Obtuse
[This object is a pull tab]
Ans
wer
C ScaleneF Right
![Page 73: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/73.jpg)
Slide 48 / 210
26 Classify the triangle by sides and angles
A EquilateralB IsoscelesC ScaleneD AcuteE EquiangularF RightG Obtuse
AB
120°
C
![Page 74: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/74.jpg)
Slide 48 (Answer) / 210
26 Classify the triangle by sides and angles
A EquilateralB IsoscelesC ScaleneD AcuteE EquiangularF RightG Obtuse
AB
120°
C[This object is a pull tab]
Ans
wer C Scalene
G Obtuse
![Page 75: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/75.jpg)
Slide 49 / 210
L
MN
27 Classify the triangle by sides and angles
A EquilateralB IsoscelesC ScaleneD AcuteE EquiangularF RightG Obtuse
![Page 76: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/76.jpg)
Slide 49 (Answer) / 210
L
MN
27 Classify the triangle by sides and angles
A EquilateralB IsoscelesC ScaleneD AcuteE EquiangularF RightG Obtuse
[This object is a pull tab]
Ans
wer
B IsoscelesF Right
![Page 77: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/77.jpg)
Slide 50 / 210
H
J
K45°
85°
50°
28 Classify the triangle by sides and angles
A EquilateralB IsoscelesC ScaleneD AcuteE EquiangularF RightG Obtuse
![Page 78: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/78.jpg)
Slide 50 (Answer) / 210
H
J
K45°
85°
50°
28 Classify the triangle by sides and angles
A EquilateralB IsoscelesC ScaleneD AcuteE EquiangularF RightG Obtuse
[This object is a pull tab]
Ans
wer C Scalene
D Acute
![Page 80: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/80.jpg)
Slide 52 / 210
Triangle Sum Theorem
A
B C
We can use what we learned about parallel lines to determine the sum of the measures of the angles of any triangle.
First, let's draw two parallel lines. The first along the base of the triangle and the other through the opposite vertex.
![Page 81: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/81.jpg)
Slide 53 / 210
And extend AB to make it a transversal.
Then, let's label some of the angles.
Triangle Sum Theorem
A
B C
x
x
y
y
![Page 82: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/82.jpg)
Slide 54 / 210
29 What is the name for the pair of angles labeled x and what is the relationship between them?
A outside exterior, they are unequalB alternate interior, they are unequalC alternate interior, they are equalD outside exterior, they are equal
Is the same true for the pair of angles labeled y?
![Page 83: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/83.jpg)
Slide 54 (Answer) / 210
29 What is the name for the pair of angles labeled x and what is the relationship between them?
A outside exterior, they are unequalB alternate interior, they are unequalC alternate interior, they are equalD outside exterior, they are equal
Is the same true for the pair of angles labeled y?[This object is a pull tab]
Ans
wer
C; yes
![Page 84: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/84.jpg)
Slide 55 / 210
A
B C
Therefore, both angles labeled x are equal and can be called x, and x has the same measure as B.
x
x
Repeat the same process with side AC and find an angle along the upper parallel line equal to angle C
Triangle Sum Theorem
![Page 85: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/85.jpg)
Slide 56 / 210
A
B C
x
x
y
y
Let's just re-label the upper angles with A, B and C.
Triangle Sum Theorem
![Page 86: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/86.jpg)
Slide 57 / 210
A
B C
The sum of those angles along that upper parallel line equals 180º, so A + B + C = 180º
B C
We made no special assumptions about this triangle, so this proof applies to all triangles: the sum of the interior angles of any triangle is 180º
Triangle Sum Theorem
![Page 87: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/87.jpg)
Slide 58 / 210
The measures of the interior angles of a triangle sum to 180°
Click here to go to the lab titled, "Triangle Sum Theorem"
Triangle Sum Theorem
A
B C
![Page 88: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/88.jpg)
Slide 59 / 210
Example: Triangle Sum Theorem
320
J
K L 200
Find the measure of the missing angle.
![Page 89: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/89.jpg)
Slide 59 (Answer) / 210
Example: Triangle Sum Theorem
320
J
K L 200
Find the measure of the missing angle.
[This object is a pull tab]
Ans
wer
m∠J + m∠K + m∠L = 180º
32º + m∠K + 20 = 180º
m∠K + 52º = 180º
m∠K = 128º
![Page 90: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/90.jpg)
Slide 60 / 210
30 What is m∠B?
A B
C
52°
53°
![Page 91: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/91.jpg)
Slide 60 (Answer) / 210
30 What is m∠B?
A B
C
52°
53°[This object is a pull tab]
Ans
wer 52 + 53 + m∠B = 180°
105 + m∠B = 180° -105 -105 m∠B = 75°
![Page 92: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/92.jpg)
Slide 61 / 210
31 What is the measurement of the missing angle?
57°L
M
N
![Page 93: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/93.jpg)
Slide 61 (Answer) / 210
31 What is the measurement of the missing angle?
57°L
M
N
[This object is a pull tab]
Ans
wer
90 + 57 + m∠N = 180° 147 + m∠N = 180° -147 -147 m∠N = 33°
![Page 94: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/94.jpg)
Slide 62 / 210
32 In ΔABC, if m∠B is 84° and m∠C is 36°, what is m∠A?
![Page 95: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/95.jpg)
Slide 62 (Answer) / 210
32 In ΔABC, if m∠B is 84° and m∠C is 36°, what is m∠A?
[This object is a pull tab]
Ans
wer
triangle ABC = 180°So, 84 + 36 + m∠A = 180°
m∠A =180° - 120°m∠A = 60°
![Page 96: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/96.jpg)
Slide 63 / 210
33 In ΔDEF, if m∠D is 63° and m∠E is 12°, find m∠F.
![Page 97: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/97.jpg)
Slide 63 (Answer) / 210
33 In ΔDEF, if m∠D is 63° and m∠E is 12°, find m∠F.
[This object is a pull tab]
Ans
wer
triangle DEF = 180°So, 63 + 12 + m∠F = 180°
m∠F =180° - 75°m∠F = 105°
![Page 98: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/98.jpg)
Slide 64 / 210
Solve for x
55°
(12x+8)°
(8x-3)°P
Q
R
Example
![Page 99: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/99.jpg)
Slide 64 (Answer) / 210
Solve for x
55°
(12x+8)°
(8x-3)°P
Q
R
Example
[This object is a pull tab]
Ans
wer m∠P + m∠Q + m∠R = 180º
55º + (12x+8) + (8x-3) = 180º20x +60 = 18020x = 120x = 6
![Page 100: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/100.jpg)
Slide 65 / 210
Q
R
S2x° 5x°
8x°34 Solve for x.
Then find: m∠Q =
m∠ R =
m∠S =
![Page 101: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/101.jpg)
Slide 65 (Answer) / 210
Q
R
S2x° 5x°
8x°34 Solve for x.
Then find: m∠Q =
m∠ R =
m∠S =
[This object is a pull tab]
Ans
wer
2x+5x+8x = 180 15x = 180 x = 12
m∠Q = 24° m∠R = 96° m∠S = 60°
Extension Answer
![Page 102: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/102.jpg)
Slide 66 / 210
35 What is the measure of ∠B?
C
B
A
(3x-17)0
(x+40)0 (2x-5)0
![Page 103: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/103.jpg)
Slide 66 (Answer) / 210
35 What is the measure of ∠B?
C
B
A
(3x-17)0
(x+40)0 (2x-5)0[This object is a pull tab]
Ans
wer
3x-17+2x-5+x+40 = 180 6x + 18 = 180 6x = 162 x = 27 m∠B = 3(27)-17 = 81 - 17 = 64
![Page 104: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/104.jpg)
Slide 67 / 210
Corollary to Triangle Sum TheoremThe acute angles of a right triangle are complementary.
A
B
C
![Page 105: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/105.jpg)
Slide 68 / 210
Given: Triangle ABC is a right triangle
Prove: Its acute angles, Angles B and C, are complementary
A
B
C
Proof of Triangle Sum Theorem Corollary
![Page 106: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/106.jpg)
Slide 69 / 210
36 Which reason applies to step 1?A Subtraction Property of EqualityB Substitution Property of EqualityC GivenD Definition of right triangle E Definition of a right angle
A
B
C
Statement Reason1 Triangle ABC is a right triangle ?
2 Right triangles contain a right angle. ?
3 ? Interior Angles Theorem4 m∠A = 90º ?5 90º + m∠B + m∠C = 180º ?
6 m∠B + m∠C = 90º ?
7 ? Definition of complementary
Ans
wer
![Page 107: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/107.jpg)
Slide 70 / 210
37 Which reason applies to step 2?A Subtraction Property of EqualityB Substitution Property of EqualityC GivenD Definition of right triangle E Definition of a right angle
A
B
C
Statement Reason1 Triangle ABC is a right triangle ?
2 Right triangles contain a right angle. ?
3 ? Interior Angles Theorem4 m∠A = 90º ?5 90º + m∠B + m∠C = 180º ?6 m∠B + m∠C = 90º ?
7 ? Definition of complementary
Ans
wer
![Page 108: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/108.jpg)
Slide 71 / 210
38 Which reason applies to step 3?
A
B
C
A The measure of a straight angle is 180ºB m∠A + m∠B + m∠C = 180ºC m∠B + m∠C = 90ºD m∠B + m∠C = 180ºE ∠A is a right angle
Statement Reason1 Triangle ABC is a right triangle ?
2 Right triangles contain a right angle. ?
3 ? Interior Angles Theorem4 m∠A = 90º ?5 90º + m∠B + m∠C = 180º ?
6 m∠B + m∠C = 90º ?
7 ? Definition of complementary
Ans
wer
![Page 109: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/109.jpg)
Slide 72 / 210
39 Which reason applies to step 4?A Subtraction Property of EqualityB Substitution Property of EqualityC GivenD Definition of right triangle E Definition of a right angle
A
B
C
Ans
wer
Statement Reason1 Triangle ABC is a right triangle ?
2 Right triangles contain a right angle. ?
3 ? Interior Angles Theorem4 m∠A = 90º ?5 90º + m∠B + m∠C = 180º ?6 m∠B + m∠C = 90º ?
7 ? Definition of complementary
![Page 110: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/110.jpg)
Slide 73 / 210
40 Which reason applies to step 5?A Subtraction Property of EqualityB Substitution Property of EqualityC GivenD Definition of right triangle E Definition of a right angle
A
B
C
Statement Reason1 Triangle ABC is a right triangle ?
2 Right triangles contain a right angle. ?
3 ? Interior Angles Theorem4 m∠A = 90º ?5 90º + m∠B + m∠C = 180º ?
6 m∠B + m∠C = 90º ?
7 ? Definition of complementary
Ans
wer
![Page 111: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/111.jpg)
Slide 74 / 210
41 Which reason applies to step 6?A Subtraction Property of EqualityB Substitution Property of EqualityC GivenD Definition of right triangle E Definition of a right angle
A
B
C
Statement Reason1 Triangle ABC is a right triangle ?
2 Right triangles contain a right angle. ?
3 ? Interior Angles Theorem4 m∠A = 90º ?5 90º + m∠B + m∠C = 180º ?6 m∠B + m∠C = 90º ?
7 ? Definition of complementary
Ans
wer
![Page 112: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/112.jpg)
Slide 75 / 210
42 Which reason applies to step 7?
A
B
C
Statement Reason1 Triangle ABC is a right triangle ?
2 Right triangles contain a right angle. ?
3 ? Interior Angles Theorem4 m∠A = 90º ?5 90º + m∠B + m∠C = 180º ?
6 m∠B + m∠C = 90º ?
7 ? Definition of complementary
A The measure of a straight angle is 180ºB The sum of the interior angles of a
triangle is 180ºC The acute angles are complementaryD The acute angles are supplementaryE ∠A is a right angle
Ans
wer
![Page 113: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/113.jpg)
Slide 76 / 210
A
B
C
Given: Triangle ABC is a right triangle
Prove: Its acute angles, Angles B and C, are complementary
Statement Reason1 Triangle ABC is a right triangle Given
2 Right triangles contain a right angle. Definition of right triangle
3 m∠A + m∠B + m∠C = 180º Interior Angles Theorem4 m∠A = 90º Definition of right angle
5 90º + m∠B + m∠C = 180º Substitution Property of Equality
6 m∠B + m∠C = 90º Subtraction Property of Equality
7The acute angles are complementary
Definition of complementary
Proof of Triangle Sum Theorem Corollary
![Page 114: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/114.jpg)
Slide 77 / 210
Example
The measure of one acute angle of a right triangle is five times the measure of the other acute angle.
Find the measure of each acute angle.
![Page 115: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/115.jpg)
Slide 77 (Answer) / 210
Example
The measure of one acute angle of a right triangle is five times the measure of the other acute angle.
Find the measure of each acute angle.
[This object is a pull tab]
Ans
wer
x + 5x + 90 = 180º6x = 90x = 15
The measures of the angles are 15º and 75º.
![Page 116: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/116.jpg)
Slide 78 / 210
43 In a right triangle, the two acute angles sum to 90°
TrueFalse
![Page 117: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/117.jpg)
Slide 78 (Answer) / 210
43 In a right triangle, the two acute angles sum to 90°
TrueFalse
[This object is a pull tab]
Ans
wer
True
![Page 118: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/118.jpg)
Slide 79 / 210
44 What is the measurement of the missing angle?
57°L
M
N
![Page 119: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/119.jpg)
Slide 79 (Answer) / 210
44 What is the measurement of the missing angle?
57°L
M
N
[This object is a pull tab]
Ans
wer
x+57 = 90ºx = 33º
Note: we solved this problem earlier using the Triangle Sum Theorem. Use the Corollary to the Triangle Sum this time.
![Page 120: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/120.jpg)
Slide 80 / 210
45 Solve for x
A
B CCB
AWhat are the measures of the three angles?
![Page 121: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/121.jpg)
Slide 80 (Answer) / 210
45 Solve for x
A
B CCB
AWhat are the measures of the three angles?
[This object is a pull tab]
Ans
wer
3x-1+31 = 903x + 30 = 90 3x = 60 x = 20
Challenge Answerm∠A = 59ºm∠B = 90ºm∠C = 31º
![Page 122: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/122.jpg)
Slide 81 / 210
46 Solve for x
What are the measures of the three angles?
![Page 123: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/123.jpg)
Slide 81 (Answer) / 210
46 Solve for x
What are the measures of the three angles?
[This object is a pull tab]
Ans
wer
2x-2+x+5 = 903x + 3 = 90 3x = 87 x = 29
Challenge Answerm∠D = 90ºm∠E = 56ºm∠F = 34º
![Page 124: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/124.jpg)
Slide 82 / 210
47 m∠1 + m∠2 =
1
23
![Page 125: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/125.jpg)
Slide 82 (Answer) / 210
47 m∠1 + m∠2 =
1
23
[This object is a pull tab]
Ans
wer
90
![Page 126: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/126.jpg)
Slide 83 / 210
48 m∠1 + m∠3 =
1
23
![Page 127: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/127.jpg)
Slide 83 (Answer) / 210
48 m∠1 + m∠3 =
1
23
[This object is a pull tab]
Ans
wer
90
![Page 128: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/128.jpg)
Slide 84 / 210
20°
X°
49 Find the value of x in the diagram
![Page 129: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/129.jpg)
Slide 84 (Answer) / 210
20°
X°
49 Find the value of x in the diagram
![Page 131: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/131.jpg)
Slide 86 / 210
Exterior angles are formed by extending any side of a triangle.
The exterior angle is then the angle between that extended side and the nearest side of the triangle.
One exterior angle is shown below.
Take a moment and draw another.
Exterior Angles
A
B Cx
![Page 132: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/132.jpg)
Slide 87 / 210
Since a triangle has three vertices and two external angles can be drawn at each vertex, it is possible to draw six external angles to a triangle.
Draw the other external angle at Vertex A.
Exterior Angles
A
B C
x
![Page 133: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/133.jpg)
Slide 88 / 210
A
B C
xx
The exterior angles at each vertex are congruent, since they are vertical angles.
Exterior Angles
![Page 134: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/134.jpg)
Slide 89 / 210
The interior angles of this triangle are ∠A, ∠ABC and ∠C.
Once an exterior angle is drawn, one interior angle is adjacent, and the two others are remote.
Since you can draw exterior angles at any vertex, any interior angle can be the remote depending on at which vertex you draw the external angle.
Remote Interior Angles
A
B Cx
In this case, ∠A and ∠C are the remote interior angles and ∠ABC is the adjacent interior angle.
![Page 135: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/135.jpg)
Slide 90 / 210
50 Which are the remote interior angles in this instance?
A ∠A & ∠BB ∠A & ∠CC ∠B & ∠C
A
B C
xx
![Page 136: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/136.jpg)
Slide 90 (Answer) / 210
50 Which are the remote interior angles in this instance?
A ∠A & ∠BB ∠A & ∠CC ∠B & ∠C
A
B C
xx
[This object is a pull tab]
Ans
wer
C
![Page 137: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/137.jpg)
Slide 91 / 210
51 If line AB is a straight line, what is the sum of ∠2 and ∠1?
1A B
2
![Page 138: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/138.jpg)
Slide 91 (Answer) / 210
51 If line AB is a straight line, what is the sum of ∠2 and ∠1?
1A B
2
[This object is a pull tab]
Ans
wer
Linear Pair PostulateSince ∠1 & ∠2 are adjacent and lie on the same line, they are a linear pair, which makes them supplementary. Since supplementary angles sum to 180º, these two angles must also sum to 180º.
In this diagram, m∠1 + m∠2 = 180º
![Page 139: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/139.jpg)
Slide 92 / 210
52 In this diagram, what is the sum of angles P, Q and R?
P
R Q
![Page 140: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/140.jpg)
Slide 92 (Answer) / 210
52 In this diagram, what is the sum of angles P, Q and R?
P
R Q
[This object is a pull tab]
Ans
wer The sum of the angles of a
triangle is 180º
![Page 141: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/141.jpg)
Slide 93 / 210
A
B Cx
D
The measure of any exterior angle of a triangle is equal to the sum of its remote interior angles.
m∠DBA = m∠A + m∠C
or
x = m∠A + m∠C
Exterior Angles Theorem
![Page 142: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/142.jpg)
Slide 94 / 210
Given: ∠DBA is an exterior angle of ΔABC and ∠A and ∠C are remote interior angles.
Prove: m∠DBA = m∠A + m∠C
Proof of Exterior Angles Theorem
A
B Cx
D
![Page 143: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/143.jpg)
Slide 95 / 210
53 Which reason applies to step 2?A Angles that form a linear pair are supplementaryB Definition of complementaryC Interior Angles TheoremD Substitution Property of EqualityE Definition of a right angle
A
B Cx
D
Statement Reason
1∠DBA is an exterior angle of ΔABC and ∠A and ∠C are remote interior angles
Given
2 ∠DBA and ∠ABC are supplementary ?
3 ? Definition of supplementary
4 m∠A+ m∠ABC + m∠C = 180° ?
5 m∠DBA + m∠ABC = m∠A + m∠ABC + m∠C ?
6 ? Subtraction Property of Equality
Ans
wer
![Page 144: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/144.jpg)
Slide 96 / 210
54 Which statement applies to step 3?A m∠DBA + m∠ABC = 180°B m∠DBA = m∠A + m∠C C m∠A + m∠B = 180° D m∠DBA + m∠A = 90°E m∠DBA + m∠A = 180°
A
B Cx
D
Statement Reason
1∠DBA is an exterior angle of ΔABC and ∠A and ∠C are remote interior angles
Given
2 ∠DBA and ∠ABC are supplementary ?
3 ? Definition of supplementary
4 m∠A+ m∠ABC + m∠C = 180° ?
5 m∠DBA + m∠ABC = m∠A + m∠ABC + m∠C ?
6 ? Subtraction Property of Equality
Ans
wer
![Page 145: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/145.jpg)
Slide 97 / 210
55 Which reason applies to step 4?A Angles that form a linear pair are supplementaryB Definition of complementaryC Interior Angles TheoremD Substitution Property of EqualityE Definition of a right angle
A
B Cx
D
Statement Reason
1∠DBA is an exterior angle of ΔABC and ∠A and ∠C are remote interior angles
Given
2 ∠DBA and ∠ABC are supplementary ?
3 ? Definition of supplementary
4 m∠A+ m∠ABC + m∠C = 180° ?
5 m∠DBA + m∠ABC = m∠A + m∠ABC + m∠C ?
6 ? Subtraction Property of Equality
Ans
wer
![Page 146: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/146.jpg)
Slide 98 / 210
56 Which reason applies to step 5?A Angles that form a linear pair are supplementaryB Definition of complementaryC Interior Angles TheoremD Substitution Property of EqualityE Definition of a right angle
A
B Cx
D
Statement Reason
1∠DBA is an exterior angle of ΔABC and ∠A and ∠C are remote interior angles
Given
2 ∠DBA and ∠ABC are supplementary ?
3 ? Definition of supplementary
4 m∠A+ m∠ABC + m∠C = 180° ?
5 m∠DBA + m∠ABC = m∠A + m∠ABC + m∠C ?
6 ? Subtraction Property of Equality
Ans
wer
![Page 147: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/147.jpg)
Slide 99 / 210
57 Which statement applies to step 6?A m∠DBA + m∠ABC = 180°B m∠DBA = m∠A + m∠C C m∠A + m∠B = 180° D m∠DBA + m∠A = 90°E m∠DBA + m∠A = 180°
A
B Cx
D
Statement Reason
1∠DBA is an exterior angle of ΔABC and ∠A and ∠C are remote interior angles
Given
2 ∠DBA and ∠ABC are supplementary ?
3 ? Definition of supplementary
4 m∠A+ m∠ABC + m∠C = 180° ?
5 m∠DBA + m∠ABC = m∠A + m∠ABC + m∠C ?
6 ? Subtraction Property of Equality
Ans
wer
![Page 148: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/148.jpg)
Slide 100 / 210
Statement Reason
1∠DBA is an exterior angle of ΔABC and ∠A and ∠C are remote interior angles
Given
2 ∠DBA and ∠ABC are supplementary Angles that form a linear pair are supplementary
3 ∠DBA + m∠ABC = 180° Definition of supplementary
4 m∠A+ m∠ABC + m∠C = 180° Interior Angles Theorem
5 m∠DBA + m∠ABC = m∠A + m∠ABC + m∠C Substitution Property of Equality
6 m∠DBA = m∠A + m∠C Subtraction Property of Equality
Proof of Exterior Angles TheoremGiven: ∠DBA is an exterior angle of ΔABC and ∠A and ∠C are remote interior angles.
Prove: m∠DBA = m∠A + m∠C
A
B Cx
D
![Page 149: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/149.jpg)
Slide 101 / 210
58 In this case, what must be the relationship between the interior angles of ΔPQR and ∠1?
A m∠Q = m∠1B m∠1 = m∠PC m∠1 = m∠Q + m∠RD m∠1 = m∠P + m∠RE m∠1 = m∠Q + m∠P
1
P
R Q
![Page 150: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/150.jpg)
Slide 101 (Answer) / 210
58 In this case, what must be the relationship between the interior angles of ΔPQR and ∠1?
A m∠Q = m∠1B m∠1 = m∠PC m∠1 = m∠Q + m∠RD m∠1 = m∠P + m∠RE m∠1 = m∠Q + m∠P
1
P
R Q[This object is a pull tab]
Ans
wer
D
![Page 151: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/151.jpg)
Slide 102 / 210
59 In this case, what must be the relationship between the interior angles of ΔPQR and ∠2?
A m∠Q = m∠2B m∠2 = m∠PC m∠2 = m∠Q + m∠RD m∠2 = m∠P + m∠RE m∠2 = m∠Q + m∠P
2P
R Q
![Page 152: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/152.jpg)
Slide 102 (Answer) / 210
59 In this case, what must be the relationship between the interior angles of ΔPQR and ∠2?
A m∠Q = m∠2B m∠2 = m∠PC m∠2 = m∠Q + m∠RD m∠2 = m∠P + m∠RE m∠2 = m∠Q + m∠P
2P
R Q
[This object is a pull tab]
Ans
wer
C
![Page 153: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/153.jpg)
Slide 103 / 210
Example: Using the Exterior Angle Theorem
140ºXº
Xº
P
QR
What is the value of x?
![Page 154: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/154.jpg)
Slide 103 (Answer) / 210
Example: Using the Exterior Angle Theorem
140ºXº
Xº
P
QR
What is the value of x?
[This object is a pull tab]
Ans
wer
140o = xo + xo
140 = 2x70 = x
The measure of the exterior angle is equal to the sum of the two angles that are not adjacent to the exterior angle.
![Page 155: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/155.jpg)
Slide 104 / 210
ExampleSolve for x and y.
21°
34°x° y°
![Page 156: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/156.jpg)
Slide 104 (Answer) / 210
ExampleSolve for x and y.
21°
34°x° y°
[This object is a pull tab]
Ans
wer
Exterior Angles Theoremx = 21 + 34 x = 55º
Triangle Sum Theorem21 + 34 + y = 180º55 + y = 180-55 -55 y = 125º
![Page 157: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/157.jpg)
Slide 105 / 210
xº yº
75º
50º
ExampleSolve for x and y.
![Page 158: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/158.jpg)
Slide 105 (Answer) / 210
xº yº
75º
50º
ExampleSolve for x and y.
[This object is a pull tab]
Ans
wer
Exterior Angles Theoremx = 75 + 50 x = 125o
Triangle Sum Theorem75 + 50 + y = 180 125 + y = 180-125 -125 y = 55o
![Page 159: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/159.jpg)
Slide 106 / 210
60 Solve for x.
xº yº
60º
55º
![Page 160: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/160.jpg)
Slide 106 (Answer) / 210
60 Solve for x.
xº yº
60º
55º[This object is a pull tab]
Ans
wer
65º
![Page 161: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/161.jpg)
Slide 107 / 210
61 Solve for y.
xºyº
60º
55º
![Page 162: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/162.jpg)
Slide 107 (Answer) / 210
61 Solve for y.
xºyº
60º
55º
[This object is a pull tab]
Ans
wer
115º
![Page 163: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/163.jpg)
Slide 108 / 210
62 Find the value of x.
2xº
yº
60º
94º
![Page 164: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/164.jpg)
Slide 108 (Answer) / 210
62 Find the value of x.
2xº
yº
60º
94º
[This object is a pull tab]
Ans
wer 60 + 2x = 94º
2x = 34x = 17
![Page 165: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/165.jpg)
Slide 109 / 210
63 Find the value of x.
(2x+3)º
yº100º
51º
![Page 166: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/166.jpg)
Slide 109 (Answer) / 210
63 Find the value of x.
(2x+3)º
yº100º
51º
[This object is a pull tab]
Ans
wer 51 + 2x + 3 = 100º2x + 54 = 1002x = 46x = 23
![Page 167: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/167.jpg)
Slide 110 / 210
64 Find the value of x.
(x+2)°
y°(3x-5)°
33°
![Page 168: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/168.jpg)
Slide 110 (Answer) / 210
64 Find the value of x.
(x+2)°
y°(3x-5)°
33°[This object is a pull tab]
Ans
wer
33 + x + 2 = 3x - 5x + 35 = 3x - 535 = 2x - 540 = 2x20 = x
![Page 169: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/169.jpg)
Slide 111 / 210
65 Segment PS bisects ∠RST, what is the value of w?
w
25°
P
S
TR
![Page 170: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/170.jpg)
Slide 111 (Answer) / 210
65 Segment PS bisects ∠RST, what is the value of w?
w
25°
P
S
TR
[This object is a pull tab]
Ans
wer
25 o115oPR T
25o
65o
S
![Page 171: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/171.jpg)
Slide 112 / 210
Example
Find the missing angles in the diagram.
60°
7
103°
43°45°
30°
5 43
2 1
![Page 172: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/172.jpg)
Slide 112 (Answer) / 210
Example
Find the missing angles in the diagram.
60°
7
103°
43°45°
30°
5 43
2 1
[This object is a pull tab]
Teac
her N
ote
Find the measures of all angles togetherm∠1 = 45o
m∠2 = 90o
m∠3 = 60o
m∠4 = 60o
m∠5 = 77o
m∠6 = 77o
m∠7 = 43o
![Page 173: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/173.jpg)
Slide 113 / 210
40º
1
24 53
60º
66 Find the measure of ∠1.
![Page 174: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/174.jpg)
Slide 113 (Answer) / 210
40º
1
24 53
60º
66 Find the measure of ∠1.
[This object is a pull tab]
Ans
wer
40 + m∠1 = 90ºm∠1 = 50º
![Page 175: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/175.jpg)
Slide 114 / 210
67 Find the measure of ∠2.
40º
1
24 53
60º
![Page 176: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/176.jpg)
Slide 114 (Answer) / 210
67 Find the measure of ∠2.
40º
1
24 53
60º
[This object is a pull tab]
Ans
wer
40 + m∠2 = 180ºm∠2 = 140º
![Page 177: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/177.jpg)
Slide 115 / 210
68 Find the measure of ∠3.
40º
1
24 53
60º
![Page 178: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/178.jpg)
Slide 115 (Answer) / 210
68 Find the measure of ∠3.
40º
1
24 53
60º
[This object is a pull tab]
Ans
wer
40 + m∠3 = 180ºm∠3 = 140º
![Page 179: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/179.jpg)
Slide 116 / 210
69 Find the measure of ∠4.
40º
1
24 53
60º
![Page 180: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/180.jpg)
Slide 116 (Answer) / 210
69 Find the measure of ∠4.
40º
1
24 53
60º
[This object is a pull tab]
Ans
wer m∠4 = 40º
vertical angles are congruent
![Page 181: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/181.jpg)
Slide 117 / 210
70 Find the measure of ∠5.
40º
1
24 53
60º
![Page 182: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/182.jpg)
Slide 117 (Answer) / 210
70 Find the measure of ∠5.
40º
1
24 53
60º
[This object is a pull tab]
Ans
wer
m∠4 = 40º
Interior angles of a triangle add up to 180º, so40+60+m∠5 = 180º100+m∠5 = 180ºm∠5 = 80o
![Page 184: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/184.jpg)
Slide 119 / 210
Inequalities in one Triangle
To investigate inequalities in one triangle download the sketch, "inequalities in one triangle" and the worksheet, "inequalities in one triangle"
Go to the sketch, "Inequalities in one
triangle."
Go to the worksheet,"Inequalities in one triangle."
![Page 185: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/185.jpg)
Slide 120 / 210
Angle Inequalities in a Triangle
The longest side is always opposite the largest angle.
The shortest side is always opposite the smallest angle.
![Page 186: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/186.jpg)
Slide 121 / 210
71 Name the longest side of this triangle.A ABB BCC CAD They are all equal
AB
C
35°60°
85°
![Page 187: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/187.jpg)
Slide 121 (Answer) / 210
71 Name the longest side of this triangle.A ABB BCC CAD They are all equal
AB
C
35°60°
85°
[This object is a pull tab]
Ans
wer
A
![Page 188: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/188.jpg)
Slide 122 / 210
72 Name the shortest side of this triangle.A ABB BCC CAD They are all equal
AB
C
35°60°
85°
![Page 189: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/189.jpg)
Slide 122 (Answer) / 210
72 Name the shortest side of this triangle.A ABB BCC CAD They are all equal
AB
C
35°60°
85°
[This object is a pull tab]
Ans
wer
B
![Page 190: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/190.jpg)
Slide 123 / 210
73 Name the shortest side of this triangle.A ABB BCC CAD They are all equal
AB
C
35° 105°
40°
![Page 191: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/191.jpg)
Slide 123 (Answer) / 210
73 Name the shortest side of this triangle.A ABB BCC CAD They are all equal
AB
C
35° 105°
40°
[This object is a pull tab]
Ans
wer
B
![Page 192: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/192.jpg)
Slide 124 / 210
74 Name the largest angle of this triangle.ABCD They are all equal
AB
C
10
148
![Page 193: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/193.jpg)
Slide 124 (Answer) / 210
74 Name the largest angle of this triangle.ABCD They are all equal
AB
C
10
148 [This object is a pull
tab]
Ans
wer
B
![Page 194: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/194.jpg)
Slide 125 / 210
75 Name the smallest angle of this triangle.ABCD They are all equal
AB
C
10
148
![Page 195: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/195.jpg)
Slide 125 (Answer) / 210
75 Name the smallest angle of this triangle.ABCD They are all equal
AB
C
10
148 [This object is a pull
tab]
Ans
wer
A
![Page 196: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/196.jpg)
Slide 126 / 210
A
C
1010
10
76 Name the smallest angle of this triangle.ABCD They are all equal
B
![Page 197: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/197.jpg)
Slide 126 (Answer) / 210
A
C
1010
10
76 Name the smallest angle of this triangle.ABCD They are all equal
B [This object is a pull tab]
Ans
wer
D
![Page 198: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/198.jpg)
Slide 127 / 210
Length Inequalities in a Triangle
No side can be longer than the sum of the other two sides.
No side can be less than the difference of the other two sides.
![Page 199: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/199.jpg)
Slide 128 / 210
Length Inequalities in a Triangle
No side can be longer than the sum of the other two sides.
This follows from the fact that if the two shorter sides cannot be placed at a 180º angle and exceed the length of the longest side, a triangle cannot be made.
As shown below, if the blue side is longer than the sum of the red and the green side, it cannot form a triangle.
Move the sides below and try to form a triangle.
![Page 200: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/200.jpg)
Slide 129 / 210
Length Inequalities in a TriangleNo side can be less than the difference of the other two sides.
This follows from the fact that if the longer sides cannot, when placed at a 0° angle, reach the end of the shorter side, a triangle cannot be made.
As shown below, if the blue side is too short to reach the red line, even when the red line is at the smallest angle, it cannot form a triangle.
![Page 201: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/201.jpg)
Slide 130 / 210
77 What is the maximum length of the third side to form a triangle if the other sides are 4 and 6?
![Page 202: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/202.jpg)
Slide 130 (Answer) / 210
77 What is the maximum length of the third side to form a triangle if the other sides are 4 and 6?
[This object is a pull tab]
Ans
wer
third side < 4 + 6third side < 10
![Page 203: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/203.jpg)
Slide 131 / 210
78 What is the maximum length of the third side to form a triangle if the other sides are 8 and 7?
![Page 204: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/204.jpg)
Slide 131 (Answer) / 210
78 What is the maximum length of the third side to form a triangle if the other sides are 8 and 7?
[This object is a pull tab]
Ans
wer third side < 8 + 7
third side < 15
![Page 205: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/205.jpg)
Slide 132 / 210
79 What is the minimum length of the third side to form a triangle if the other sides are 4 and 6?
![Page 206: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/206.jpg)
Slide 132 (Answer) / 210
79 What is the minimum length of the third side to form a triangle if the other sides are 4 and 6?
[This object is a pull tab]
Ans
wer 6 < third side + 4
2 < third side
![Page 207: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/207.jpg)
Slide 133 / 210
80 What is the minimum length of the third side to form a triangle if the other sides are 7 and 8?
![Page 208: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/208.jpg)
Slide 133 (Answer) / 210
80 What is the minimum length of the third side to form a triangle if the other sides are 7 and 8?
[This object is a pull tab]
Ans
wer 8 < third side + 7
1 < third side
![Page 210: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/210.jpg)
Slide 135 / 210
Recall that:
Congruence
Two objects are congruent if they can be moved, by any combination of translation, rotation and reflection, so that every part of each object overlaps.
This is the symbol for congruence:
If a is congruent to b, this would be shown as
which is read as "a is congruent to b."
a b
![Page 211: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/211.jpg)
Slide 136 / 210
Only line segments with the same length are congruent.
Also, all congruent segments have the same length.
We learned earlier that:
Congruent Line Segments
ab
cd
c da b
![Page 212: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/212.jpg)
Slide 137 / 210
Recall:
Congruent Angles
A B∠ ∠ ∠C ∠D
Two angles are congruent if they have the same measure.
Two angles are not congruent if they have different measures.
AB
C
D
If m∠A = m∠B If m∠C # m∠D
![Page 213: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/213.jpg)
Slide 138 / 210
Congruent Triangles
Triangles are made up of
three line segments AND three angles
For one triangle to be congruent to another
all three sides AND all three angles must be congruent.
![Page 214: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/214.jpg)
Slide 139 / 210
Similar Triangles
If all the sides of two triangles are congruent, we will soon show that all the angles are also congruent.
Therefore, the triangles are congruent.
However, two triangles can have all their angles congruent, with all or none of their sides being congruent.
In that case, they are said to be Similar Triangles.
![Page 215: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/215.jpg)
Slide 140 / 210
Congruent Triangles
Congruent Triangles are also Similar Triangles since their angles are all congruent.
Congruent triangles are therefore a special case of similar triangles.We will focus on similar triangles first, and then work
with congruent triangles in a later unit.
Similar triangles represent a great tool to solve problems, and are the foundation of trigonometry.
![Page 216: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/216.jpg)
Slide 141 / 210
Similar triangles have the same shape, but can have different sizes.
If they have the same shape and are the same size, they are both similar and congruent.
A
B
C D
E
F
Similar Triangles Have Proportional Sides Theorem
![Page 217: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/217.jpg)
Slide 142 / 210
Similar Triangles
This is the symbol for similarity
So, the symbolic statement for
Triangle ABC is similar to Triangle DEF
is:
DEFDEFΔABC Δ
![Page 218: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/218.jpg)
Slide 143 / 210
Naming Similar Triangles
This statement tells you more than that the triangles are similar.
It also tells you which angles are equal.
In this case, that
m∠A = m∠D m∠B = m∠E m∠C = m∠F
And, thereby which are the corresponding, proportional, sides.
AB corresponds to DEBC corresponds to EFCA corresponds to FD
DEFDEFΔABC Δ
![Page 219: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/219.jpg)
Slide 144 / 210
Naming Similar Triangles
So, when you are naming similar triangles, the order of the letters matters.
They don't have to be alphabetical.
But they have to be named so that equal angles correspond to one another.
DEFDEFΔABC Δ
![Page 220: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/220.jpg)
Slide 145 / 210
Proving Triangles Similar
If you can prove that all three angles of two triangles are congruent, you have directly proven that they are similar.
However, there are shortcuts to proving triangles similar.
We will explore three sets of conditions that imply the three angles of two triangles are congruent, meaning that the triangles must be similar.
![Page 221: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/221.jpg)
Slide 146 / 210
Angle-Angle Similarity Theorem
We know from the Triangle Sum Theorem that the sum of the interior angles of a triangle is always 180o.
So, if two triangles have two pair of congruent angles which sum to x, then the third angle in both triangles must be (180o - x) ....forming three congruent pairs of angles.
One way to prove that two triangles are similar is to prove that two of the angles in each triangle are
congruent.
![Page 222: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/222.jpg)
Slide 147 / 210
Angle-Angle Similarity TheoremIf two angles of a triangle are congruent to two angles of another triangle, their third angles are congruent and the triangles are similar.
Here's the proof:Statement Reason
1 ∠A and ∠B in ΔABC are ≅ to ∠D and ∠E in ΔDEF
Given
2 m∠A = m∠D; m∠B = m∠E Definition of Congruent Angles
3 m∠A+ m∠B + m∠C = 180ºm∠D+ m∠E + m∠F = 180º
Triangle Sum Theorem
4 m∠C =180º - (m∠A + m∠B) m∠F =180º- (m∠D + m∠E)
Subtraction Property of Equality
5 m∠C =180º - (m∠A + m∠B) m∠F =180º- (m∠A + m∠B)
Substitution Property of Equality
6 m∠C = m∠F Substitution Property of Equality
7 ΔABC and ΔDEF are similar Definition of Similarity
![Page 223: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/223.jpg)
Slide 148 / 210
If two triangles have their sides proportional, the triangles will be equiangular and will have those angles equal which their corresponding sides subtend.
Euclid - Book Six: Proposition 5
Equiangular triangles are similar, so this states that triangles with proportional sides are similar.
This is a second way to prove triangles are similar:
If you can prove that all three pairs of sides in two triangles are proportional, then you have proven the
triangles similar.
Side-Side-Side Similarity Theorem
![Page 224: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/224.jpg)
Slide 149 / 210
Side-Side-Side Similarity TheoremThis follows from the way we constructed congruent angles.
We made use of the fact that if angles are congruent, their sides are separating at the same rate as you move away from the vertex. Here's the drawing we used to construct ∠ABC so it would be congruent to ∠FGH.
F
G H
A
CB
![Page 225: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/225.jpg)
Slide 150 / 210
Side-Side-Side Similarity TheoremIf we draw the green line segments connecting the points where the blue arcs intersect the rays, we can see that the length of that segment would be the same for both angles.
Since the angles are congruent, the line segment opposite those angles will also be congruent, if it intersects both sides of the angle at the same distance from the vertex in both cases.
F
G H
A
CB
D
E
![Page 226: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/226.jpg)
Slide 151 / 210
Side-Side-Side Similarity TheoremIn this case the segments AC and DE will be congruent since segments GD and GE are also congruent to segments AB and BC.
Therefore ΔDEG is congruent to ΔABC, since all the sides and angles are the same.
Changing the scale of ΔABC won't change the angle measures. The sides would then be in proportion to those of ΔDEG, but not equal.
F
G H
A
CB
D
E
![Page 227: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/227.jpg)
Slide 152 / 210
A
CB
Side-Side-Side Similarity TheoremThe diagram below shows an expansion of ΔABC and we see that the measures of the angles are unchanged.
They are still similar triangles. The corresponding sides are in proportion.
F
G H
D
E
![Page 228: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/228.jpg)
Slide 153 / 210
A
CB
Side-Side-Side Similarity TheoremRemoving the arcs and shifting the smaller triangle within the larger makes it clear that all angles are congruent and the sides are in proportion.
So, the second way to prove triangles similar is to show that all their sides are in proportion.
F
D
EG H
![Page 229: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/229.jpg)
Slide 154 / 210
Side-Angle-Side Similarity Theorem
If two triangles have one angle equal to another and the sides about the equal angle are in proportion, the triangles will be equiangular and will have those angles equal which the corresponding sides subtend.
Euclid's Elements - Book Six: Proposition 6
The third way to prove triangles are similar is to show they share an angle which is equal and the two sides forming that angle are proportional in the two triangles.
![Page 230: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/230.jpg)
Slide 155 / 210
Side-Angle-Side Similarity Theorem
This directly follows from the work we just did to show that Side-Side-Side proportionality can be used to prove triangles are similar.
If you recall, the line segment which makes up the third side of a triangle is completely defined by its opposite angle and the lengths of the other two sides.
![Page 231: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/231.jpg)
Slide 156 / 210
Side-Angle-Side Similarity Theorem
If the angles are congruent and the two sides of the angle are in proportion, the third side must also be in proportion.
If all three sides are in proportion, the triangles must be similar due to the Side-Side-Side Theorem.
You can see that on the next page.
![Page 232: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/232.jpg)
Slide 157 / 210
A
B
C D
E
F
Side-Angle Side Similarity TheoremIf ∠B ≅ ∠E and segments AB and BC are proportional to segments ED and EF, then segment AC must also be proportional to segment DF. Since all the sides are in proportion, the triangles are similar.
![Page 233: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/233.jpg)
Slide 158 / 210
Common Error
You CANNOT prove triangles similar using Side-Side-Angle.
This is not the same as Side-Angle-Side.
As shown below, two triangles can have two corresponding sides and one corresponding angle congruent, but NOT be similar.
![Page 234: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/234.jpg)
Slide 159 / 210
81 Which theorem allows you to prove these two triangles are similar?
A Angle-AngleB Side-Angle-SideC Side-Side-SideD They may not be similar
x
x
E They are not similar
![Page 235: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/235.jpg)
Slide 159 (Answer) / 210
81 Which theorem allows you to prove these two triangles are similar?
A Angle-AngleB Side-Angle-SideC Side-Side-SideD They may not be similar
x
x
E They are not similar
[This object is a pull tab]
Ans
wer A
x
x
![Page 236: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/236.jpg)
Slide 160 / 210
82 Which theorem allows you to prove these two triangles are similar?
A Angle-AngleB Side-Angle-SideC Side-Side-SideD They may not be similarE They are not similar
![Page 237: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/237.jpg)
Slide 160 (Answer) / 210
82 Which theorem allows you to prove these two triangles are similar?
A Angle-AngleB Side-Angle-SideC Side-Side-SideD They may not be similarE They are not similar
[This object is a pull tab]
Ans
werD; not enough information
![Page 238: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/238.jpg)
Slide 161 / 210
83 Which theorem allows you to prove these two triangles are similar?
A Angle-AngleB Side-Angle-SideC Side-Side-Side
D They may not be similar
6
4
88
12
E They are not similar
16
![Page 239: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/239.jpg)
Slide 161 (Answer) / 210
83 Which theorem allows you to prove these two triangles are similar?
A Angle-AngleB Side-Angle-SideC Side-Side-Side
D They may not be similar
6
4
88
12
E They are not similar
16[This object is a pull tab]
Ans
wer
C
![Page 240: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/240.jpg)
Slide 162 / 210
84 Which theorem allows you to prove these two triangles are similar?
A Angle-AngleB Side-Angle-SideC Side-Side-SideD They may not be similar
4 8
36
6 10
E They are not similar
![Page 241: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/241.jpg)
Slide 162 (Answer) / 210
84 Which theorem allows you to prove these two triangles are similar?
A Angle-AngleB Side-Angle-SideC Side-Side-SideD They may not be similar
4 8
36
6 10
E They are not similar
[This object is a pull tab]
Ans
wer
E; the ratios are not the same
![Page 242: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/242.jpg)
Slide 163 / 210
85 Which theorem allows you to prove these two triangles are similar?
4 8
36
xx
A Angle-AngleB Side-Angle-SideC Side-Side-SideD They may not be similarE They are not similar
![Page 243: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/243.jpg)
Slide 163 (Answer) / 210
85 Which theorem allows you to prove these two triangles are similar?
4 8
36
xx
A Angle-AngleB Side-Angle-SideC Side-Side-SideD They may not be similarE They are not similar
[This object is a pull tab]
Ans
wer
B
![Page 244: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/244.jpg)
Slide 164 / 210
86 Which theorem allows you to prove these two triangles are similar?
A Angle-AngleB Side-Angle-SideC Side-Side-SideD They may not be similar
4
3x
8
6
xE They are not similar
![Page 245: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/245.jpg)
Slide 164 (Answer) / 210
86 Which theorem allows you to prove these two triangles are similar?
A Angle-AngleB Side-Angle-SideC Side-Side-SideD They may not be similar
4
3x
8
6
xE They are not similar
[This object is a pull tab]
Ans
wer
D
4
3x
8
6
x
These are NOT similar
4
3x
8
6
x
These are similar
x
x
![Page 246: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/246.jpg)
Slide 165 / 210
87 Which theorem allows you to prove these two triangles are similar?
A Angle-AngleB Side-Angle-SideC Side-Side-SideD They may not be similarE They are not similar
![Page 247: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/247.jpg)
Slide 165 (Answer) / 210
87 Which theorem allows you to prove these two triangles are similar?
A Angle-AngleB Side-Angle-SideC Side-Side-SideD They may not be similarE They are not similar
[This object is a pull tab]
Ans
wer
C
![Page 248: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/248.jpg)
Slide 166 / 210
88 Which theorem allows you to prove these two triangles are similar?
A Angle-AngleB Side-Angle-SideC Side-Side-SideD They may not be similarE They are not similar
![Page 249: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/249.jpg)
Slide 166 (Answer) / 210
88 Which theorem allows you to prove these two triangles are similar?
A Angle-AngleB Side-Angle-SideC Side-Side-SideD They may not be similarE They are not similar
[This object is a pull tab]
Ans
wer
D
x x
x y
These are similar
These are NOT similar
![Page 250: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/250.jpg)
Slide 167 / 210
89 Which theorem allows you to prove these two triangles are similar?
A Angle-AngleB Side-Angle-SideC Side-Side-SideD They may not be similarE They are not similar
![Page 251: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/251.jpg)
Slide 167 (Answer) / 210
89 Which theorem allows you to prove these two triangles are similar?
A Angle-AngleB Side-Angle-SideC Side-Side-SideD They may not be similarE They are not similar
[This object is a pull tab]
Ans
wer
B
![Page 252: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/252.jpg)
Slide 168 / 210
90 Which theorem allows you to prove these two triangles are similar?
A Angle-AngleB Side-Angle-SideC Side-Side-SideD They may not be similarE They are not similar
A
B C
D ENote that BC is parallel to DE.
![Page 253: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/253.jpg)
Slide 168 (Answer) / 210
90 Which theorem allows you to prove these two triangles are similar?
A Angle-AngleB Side-Angle-SideC Side-Side-SideD They may not be similarE They are not similar
A
B C
D ENote that BC is parallel to DE.
[This object is a pull tab]
Ans
wer
A; using any 2 of the 3 angles marked in diagram
A
B C
D E
![Page 254: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/254.jpg)
Slide 169 / 210
A
B C
D E
Side Splitter Theorem
Any line parallel to a side of a triangle will form a triangle which is similar to the first triangle.
As we will learn later, it also makes all the sides proportional, splitting them...hence the name of the theorem.
![Page 255: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/255.jpg)
Slide 170 / 210
A
B C
D E
Proof of Side Splitter Theorem
Given: BC is parallel to DE
Prove: ΔABC ~ ΔADE.
![Page 256: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/256.jpg)
Slide 171 / 210
91 What is the reason for step 2?
A Angle-Angle Similarity TheoremB Side-Side-Side Similarity TheoremC Reflexive Property of Congruence D When two parallel lines are intersected by a transversal,
the corresponding angles are congruent.E When two parallel lines are intersected by a transversal,
the alternate interior angles are congruent.
A
B C
D E
Ans
wer
Statement Reason1 BC is parallel to DE Given2 ∠ABC ≅ ∠D; ∠ACB ≅ ∠E ?3 ∠A ≅ ∠A ?4 ΔABC ~ ΔADE ?
![Page 257: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/257.jpg)
Slide 172 / 210
92 What is the reason for step 3?
A Angle-Angle Similarity TheoremB Side-Side-Side Similarity TheoremC Reflexive Property of Congruence D When two parallel lines are intersected by a transversal,
the corresponding angles are congruent.E When two parallel lines are intersected by a transversal,
the alternate interior angles are congruent.
Ans
wer
A
B C
D E
Statement Reason1 BC is parallel to DE Given2 ∠ABC ≅ ∠D; ∠ACB ≅ ∠E ?3 ∠A ≅ ∠A ?4 ΔABC ~ ΔADE ?
![Page 258: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/258.jpg)
Slide 173 / 210
93 What is the reason for step 4?
A Angle-Angle Similarity Theorem
B Side-Side-Side Similarity Theorem
C Reflexive Property of Congruence D When two parallel lines are intersected by a transversal,
the corresponding angles are congruent.E When two parallel lines are intersected by a transversal,
the alternate interior angles are congruent.
A
B C
D E
Statement Reason1 BC is parallel to DE Given2 ∠ABC ≅ ∠D; ∠ACB ≅ ∠E ?3 ∠A ≅ ∠A ?4 ΔABC ~ ΔADE ?
Ans
wer
![Page 259: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/259.jpg)
Slide 174 / 210
Proof of Side Splitter Theorem
Given: BC is parallel to DE
Prove: ΔABC ~ ΔADE
A
B C
D EStatement Reason
1 BC is parallel to DE Given
2 ∠ABC ≅ ∠D; ∠ACB ≅ ∠EWhen two parallel lines are
intersected by a transvesal, the corresponding angles are congruent
3 ∠A ≅ ∠A Reflexive Property of Congruence
4 ΔABC ~ ΔADE Angle-Angle Similarity Theorem
![Page 260: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/260.jpg)
Slide 175 / 210
Similar triangles have the same shape, but can have different sizes. If they have the same shape and are the same size, they are congruent.
If they have the same shape and are different sizes, they are similar and their sides are in proportion.
A
B
C D
E
F
Similar Triangles Have Proportional Sides Theorem
![Page 261: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/261.jpg)
Slide 176 / 210
The converse is also true, and will prove very useful.
If two triangles are similar, all of their corresponding sides are in proportion.
*While Euclid does prove this theorem, his proof relies on other theorems which would have to be proven first and would take us beyond the scope of this course. So, we'll just rely on this theorem and note that the proof is available in The Elements by Euclid - Book Six: Proposition 5.
Similar Triangles Have Proportional Sides Theorem
![Page 262: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/262.jpg)
Slide 177 / 210
Similar Triangles and Proportionality
A
B
C D
E
F
In the triangles below, if we know that
m∠A = m∠D, m∠B = m∠E, and m∠C = m∠F,
then we know that the triangles are similar.
![Page 263: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/263.jpg)
Slide 178 / 210
Similar Triangles and Proportionality
A
B
C D
E
F
We also then know that the corresponding sides are proportional.
The symbol for proportional is the Greek letter, alpha: #
AB α DE, since AB corresponds to DEBC α EF, since BC corresponds to EFAC α DF, since AC corresponds to DF
![Page 264: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/264.jpg)
Slide 179 / 210
Corresponding Sides
A
B
C D
E
F
Our work with similar triangles and our future work with congruent triangles requires us to identify the corresponding sides.
One way to do that is to locate the sides opposite congruent angles. If we know triangles ABC and EDF are similar and that angle A is congruent to angle D, then the sides opposite A and D are in proportion: BC α EF
![Page 265: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/265.jpg)
Slide 180 / 210
Corresponding Sides
A
B
C D
E
F
Another way of identifying corresponding sides is to use Euclid's description "...those angles [are] equal which their corresponding sides subtend."
Below, since angle A is equal to angle D and angle B is equal to angle E, then sides AB and DE are in proportion.
![Page 266: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/266.jpg)
Slide 181 / 210
Corresponding Sides
A
B
C D
E
F
Either approach works; use the one you find easiest.
Identify corresponding sides as the sides connecting equal angles or the sides opposite equal angles...you'll get the same result.
![Page 267: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/267.jpg)
Slide 182 / 210
Similar Triangles and Proportionality
A
B
C D
E
F
Another way of saying two sides are proportional is to say that one is a scaled-up version of the other. If you multiply all the sides of one triangle by the same scale factor, k, you get the other triangle. In this case, if ΔABC is k times as big as ΔDEF, then: AB = kDE BC = kEF AC = kDF
![Page 268: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/268.jpg)
Slide 183 / 210
Similar Triangles and Proportionality
A
B
C D
E
F
Or, dividing the corresponding sides yields:
AB BC ACDE EF DF = k= =
This property of proportionality is very useful in solving problems using similar triangles, and provides the foundation for trigonometry.
![Page 269: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/269.jpg)
Slide 184 / 210
94 If m∠A = m∠D, m∠B = m∠E, and m∠C = m∠F, identify which side corresponds to side AB.
A DEB EFC FG
A
B
C D
E
F
![Page 270: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/270.jpg)
Slide 184 (Answer) / 210
94 If m∠A = m∠D, m∠B = m∠E, and m∠C = m∠F, identify which side corresponds to side AB.
A DEB EFC FG
A
B
C D
E
F
[This object is a pull tab]
Ans
wer
A
![Page 271: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/271.jpg)
Slide 185 / 210
95 If m∠I = m∠M, m∠H = m∠N, and m∠J = m∠L, identify which side corresponds to side IJ.
A MNB NLC ML
I
J
H
M
N
L
![Page 272: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/272.jpg)
Slide 185 (Answer) / 210
95 If m∠I = m∠M, m∠H = m∠N, and m∠J = m∠L, identify which side corresponds to side IJ.
A MNB NLC ML
I
J
H
M
N
L
[This object is a pull tab]
Ans
wer
C
![Page 273: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/273.jpg)
Slide 186 / 210
A
B
C8 D
E
F4
Example - Proportional Sides
Given that ΔABC is similar to ΔDEF, and given the indicated lengths, find the lengths AB and BC.
5 7
![Page 274: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/274.jpg)
Slide 187 / 210
Example - Proportional Sides
Since the triangles are similar we know that the following relationship holds between all the corresponding sides.
First, let's find the constant of proportionality, k, by using the two sides for which we have values: AC and DF
AB BC ACED EF DF = k= =
A
B
C8 D
E
F4
5 7
![Page 275: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/275.jpg)
Slide 188 / 210
A
B
C
5 7
8 D
E
F4
Example - Proportional Sides
AB BC ACED EF DF = k = 2= =
AC 8 DF 4= = k = 2
That means that the other two sides of ΔABC will also be twice as large as the corresponding sides of ΔDEF
![Page 276: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/276.jpg)
Slide 189 / 210
A
B
C
5 7
8 D
E
F4
Example - Proportional Sides
AB ED = 2 BC
EF = 2
AB 5 = 2
AB = 10
BC 7 = 2
BC = 14
![Page 277: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/277.jpg)
Slide 190 / 210
96 Given that m∠A = m∠D, m∠B = m∠E, and m∠C = m∠F. If BC = 8, DE = 6, and AB = 4, EF = ?
A
B
C D
E
F
![Page 278: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/278.jpg)
Slide 190 (Answer) / 210
96 Given that m∠A = m∠D, m∠B = m∠E, and m∠C = m∠F. If BC = 8, DE = 6, and AB = 4, EF = ?
A
B
C D
E
F
[This object is a pull tab]
Ans
wer
4 6
8 x=
4x = 48 4 4x = 12 = EF
![Page 279: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/279.jpg)
Slide 191 / 210
97 Given that ΔJIH is similar to ΔLMN; find LM.
I
J
H
M
N
L
14
10
12
5
![Page 280: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/280.jpg)
Slide 191 (Answer) / 210
97 Given that ΔJIH is similar to ΔLMN; find LM.
I
J
H
M
N
L
14
10
12
5
[This object is a pull tab]
Ans
wer
12 x
10 5=
60 = 10x10 10x = 6 = LM
![Page 281: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/281.jpg)
Slide 192 / 210
98 Given that ΔJIH is similar to ΔLMN; find LN.
I
J
H
M
N
L
14
10
12
5
![Page 282: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/282.jpg)
Slide 192 (Answer) / 210
98 Given that ΔJIH is similar to ΔLMN; find LN.
I
J
H
M
N
L
14
10
12
5
[This object is a pull tab]
Ans
wer
14 x
10 5=
70 = 10x10 10x = 7 = LN
![Page 283: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/283.jpg)
Slide 193 / 210
99 Given that BC is parallel to DE and the given lengths, find DE.
A
B C
D E
8
64
![Page 284: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/284.jpg)
Slide 193 (Answer) / 210
99 Given that BC is parallel to DE and the given lengths, find DE.
A
B C
D E
8
64
[This object is a pull tab]
Ans
wer
8 12
6 x=
8x = 72 8 8x = 9 = DE
![Page 285: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/285.jpg)
Slide 194 / 210
100 Given that BC is parallel to DE and the given lengths, find DB.
A
B C
D E9
7
3
![Page 286: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/286.jpg)
Slide 194 (Answer) / 210
100 Given that BC is parallel to DE and the given lengths, find DB.
A
B C
D E9
7
3[This object is a pull tab]
Ans
wer
3 9
7 x=
3x = 63 3 3x = 21 = ADBD + 7 = 21BD = 14
![Page 287: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/287.jpg)
Slide 195 / 210
Example - Similarity & Proportional Sides
D
P
K
12
9
18
R
L
B6
1210
Determine if the triangles are similar. If they are similar write a similarity statement. If they are not similar, explain why.
![Page 288: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/288.jpg)
Slide 196 / 210
Example - Similarity & Proportional Sides
D
P
K
12
9
18
R
L
B6
1210
To identify the corresponding sides without wasting a lot of time, first list all the sides from shortest to longest of both triangles and compare to see if they are all proportional.
Then you can identify corresponding sides and the constant of proportionality.
![Page 289: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/289.jpg)
Slide 197 / 210
Example - Similarity & Proportional Sides
D
P
K
15
9
18
R
L
B6
1210
Side of ΔPDK Length Side of ΔBRL Length Ratio
DK 9 BR 6 1.5
PD 15 RL 10 1.5
PK 18 BL 12 1.5
All corresponding sides are in the ratio of 1.5:1, so the triangles are similar.
This also provides the order of the sides, so we can say that ΔKDP is similar to ΔBRL. Check to make sure that all the sides are in the correct order.
![Page 290: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/290.jpg)
Slide 198 / 210
101 If these triangles are similar, enter the constant of proportionality, k, between the larger and smaller triangle. If they are not, enter zero.
D
P
K
12
9
18
R
L
B6
1210
![Page 291: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/291.jpg)
Slide 198 (Answer) / 210
101 If these triangles are similar, enter the constant of proportionality, k, between the larger and smaller triangle. If they are not, enter zero.
D
P
K
12
9
18
R
L
B6
1210
[This object is a pull tab]
Ans
wer
DK = 9PD = 12PK = 18
BR = 6RL = 10BL = 12
9 6
12 5≠
Not similar; 0
![Page 292: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/292.jpg)
Slide 199 / 210
102 If these triangles are similar, enter the constant of proportionality, k, between the larger and smaller triangle. If they are not, enter zero.
52°
1
2
3
R
S
T
52°24
6X
Y
Z
![Page 293: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/293.jpg)
Slide 199 (Answer) / 210
102 If these triangles are similar, enter the constant of proportionality, k, between the larger and smaller triangle. If they are not, enter zero.
52°
1
2
3
R
S
T
52°24
6X
Y
Z
[This object is a pull tab]
Ans
wer
RS = 1RT = 2ST = 3
YZ = 2YX = 4ZX = 6
1 2
2 4
3 6= =
1/2 = 0.5 = k
![Page 294: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/294.jpg)
Slide 200 / 210
103 If these triangles are similar, enter the constant of proportionality, k, between the larger and smaller triangle. If they are not, enter zero.
P
R
S
3 4.2
6B
C
D
2 2.8
4
![Page 295: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/295.jpg)
Slide 200 (Answer) / 210
103 If these triangles are similar, enter the constant of proportionality, k, between the larger and smaller triangle. If they are not, enter zero.
P
R
S
3 4.2
6B
C
D
2 2.8
4
[This object is a pull tab]
Ans
wer
PR = 3RS = 4.2PS = 6
BC = 2CD = 2.8BD = 4
3 2
4.22.8
6 4= =
3/2 = 1.5 = k
![Page 296: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/296.jpg)
Slide 201 / 210
A
B C
D E
Converse of Side Splitter Theorem
If a line divides the two sides of a triangle proportionally, then the line is parallel to the third side.
![Page 297: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/297.jpg)
Slide 202 / 210
104 Find the value of x to prove that AB is parallel to ER.
27
x
18 12R
EA
B
D
![Page 298: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/298.jpg)
Slide 202 (Answer) / 210
104 Find the value of x to prove that AB is parallel to ER.
27
x
18 12R
EA
B
D [This object is a pull tab]
Ans
wer
2718
x 12=
324 = 18x 18 18 x = 18
![Page 299: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/299.jpg)
Slide 203 / 210
105 Find the value of x to prove that FC is parallel to MN.
J
M
NC
F
x 9
6
8
![Page 300: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/300.jpg)
Slide 203 (Answer) / 210
105 Find the value of x to prove that FC is parallel to MN.
J
M
NC
F
x 9
6
8
[This object is a pull tab]
Ans
wer
8 6
9 x=
8x = 54x 8 8 x = 6.75
![Page 301: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/301.jpg)
Slide 204 / 210
106 Find the value of y.
6
1012
y
![Page 302: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/302.jpg)
Slide 204 (Answer) / 210
106 Find the value of y.
6
1012
y
[This object is a pull tab]
Ans
wer
12 y
10 6=
72 = 10y10 10 y = 7.2
![Page 303: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/303.jpg)
Slide 205 / 210
107 Find the value of y.
4
14 12
y
![Page 304: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/304.jpg)
Slide 205 (Answer) / 210
107 Find the value of y.
4
14 12
y
[This object is a pull tab]
Ans
wer
14 y
12 4=
56 = 12y12 12 x = 4.6
![Page 305: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/305.jpg)
Slide 206 / 210
108 Find the value of y.
2415y
6
![Page 306: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/306.jpg)
Slide 206 (Answer) / 210
108 Find the value of y.
2415y
6
[This object is a pull tab]
Ans
wer
2124
15 y=
360 = 21y 21 21 x 17.1
![Page 308: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/308.jpg)
Slide 208 / 210
Question 1/7
![Page 309: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/309.jpg)
Slide 209 / 210
109 The figure ΔABC ~ ΔDEF with side lengths as indicated. What is the value of x?
F
D
E
95
7
C
BA
27
21
x
![Page 310: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/310.jpg)
Slide 209 (Answer) / 210
109 The figure ΔABC ~ ΔDEF with side lengths as indicated. What is the value of x?
F
D
E
95
7
C
BA
27
21
x
[This object is a pull tab]
Ans
wer
5 x
9 27 =
9x = 135
5 x
7 21 =
7x = 105OR
x = 15
![Page 311: Slide 1 / 210content.njctl.org/courses/math/archived-courses... · Slide 8 / 210 1 The letter on this triangle that corresponds to a side is: A B C. Slide 8 (Answer) / 210 1 The letter](https://reader034.vdocuments.site/reader034/viewer/2022050410/5f86c35483487532425a9198/html5/thumbnails/311.jpg)
Slide 210 / 210