properties of triangles 5
TRANSCRIPT
Ā© Carnegie Learning
451
5Properties of Triangles
5.1 Name That Triangle!
Classifying Triangles on the Coordinate Plane . . . . . . . . 453
5.2 Inside Out
Triangle Sum, Exterior Angle, and Exterior
Angle Inequality Theorems . . . . . . . . . . . . . . . . . . . . . . . 461
5.3 Trade Routes and Pasta Anyone?
The Triangle Inequality Theorem . . . . . . . . . . . . . . . . . . . 479
5.4 Stamps Around the World
Properties of a 458ā458ā908 Triangle . . . . . . . . . . . . . . . . 489
5.5 More Stamps, Really?
Properties of a 30Ā°ā60Ā°ā90Ā° Triangle . . . . . . . . . . . . . . . 497
A lot of people use email
but there is still a need to āsnailā mail too. Mail isnāt really delivered by snailsāitās just a
comment on how slow it is compared to a computer.
Ā© Carnegie Learning
451A Chapter 5 Properties of Triangles
5
Chapter 5 Overview
This chapter focuses on properties of triangles, beginning with classifying triangles on the coordinate plane. Theorems
involving angles and side lengths of triangles are presented. The last two lessons discuss properties and theorems of
45Āŗ-45Āŗ-90Āŗ triangles and 30Āŗ-60Āŗ-90Āŗ triangles.
Lesson TEKS Pacing Highlights
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5.1
Classifying
Triangles on the
Coordinate
Plane
2.B 1
This lesson provides opportunities for
students to graph and classify triangles on
the coordinate plane by their side lengths
and angle measures.
As a culminating activity, students
classify Indiaās Golden Triangle given the
citiesā coordinates.
X X
5.2
Triangle Sum,
Exterior Angle,
and Exterior
Angle Inequality
Theorems
6.D 1
In this lesson, students prove the Triangle
Sum Theorem, Exterior Angle Theorem, and
Exterior Angle Inequality Theorem.
Questions ask students to investigate the
side lengths and angle measures of triangles
before proving the theorems.
X X
5.3
The Triangle
Inequality
Theorem
5.D 1
Students complete an activity with pasta
strands to investigate the possible side
lengths that can form triangles. Students
then prove the Triangle Inequality Theorem.
X X
5.4
Properties of a
45Ā°ā45Ā°ā90Ā°
Triangle
7.A
9.B1
Students investigate the properties of
45Ā°245Ā°290Ā° triangles in this lesson.
Questions ask students to apply the
45Ā°245Ā°290Ā° Triangle Theorem and
construction to solve problems and verify
properties of 45Ā°245Ā°290Ā° triangles.
X
5.5
Properties of a
30Ā°ā60Ā°ā90Ā°
Triangle
9.B 1
Students investigate the properties of
30Ā°-60Ā°-90Ā° triangles in this lesson.
Questions ask students to apply the
30Ā°260Ā°290Ā° Triangle Theorem and
construction to solve problems and verify
properties of 30Ā°260Ā°290Ā° triangles.
As a culminating activity, students compare
the properties of 45Ā°245Ā°290Ā° triangles
with 30Ā°260Ā°290Ā° triangles.
X X
Ā© Carnegie Learning
5
Chapter 5 Properties of Triangles 451B
Skills Practice Correlation for Chapter 5
Lesson Problem Set Objectives
5.1
Classifying Triangles on the Coordinate Plane
1 ā 6Determine the possible locations of a point to create triangles on the coordinate plane given a line segment
7 ā 12Graph triangles on the coordinate plane given vertex coordinates and classify the triangles based on the side lengths
13 ā 18Graph triangles on the coordinate plane given vertex coordinates and classify the triangles based on the angle measures
5.2
Triangle Sum, Exterior Angle, and Exterior
Angle Inequality Theorems
Vocabulary
1 ā 6 Determine the measure of missing angle measures in triangles
7 ā 12 Determine the order of side lengths given information in diagrams
13 ā 18 Identify interior, exterior, and remote interior angles of triangles
19 ā 24 Solve for x given triangle diagrams
25 ā 30Write two inequalities needed to prove the Exterior Angle Inequality Theorem given triangle diagrams
5.3The Triangle Inequality Theorem
Vocabulary
1 ā 6 Order angle measures of triangles without measuring
7 ā 16 Determine whether it is possible to form a triangle from given side lengths
17 ā 22 Write inequalities to describe possible unknown side lengths of triangles
5.4Properties of a 45Ā°ā45Ā°ā90Ā° Triangle
Vocabulary
1 ā 4 Determine the length of the hypotenuse of 45Ā°245Ā°290Ā° triangles
5 ā 8 Determine the lengths of the legs of 45Ā°245Ā°290Ā° triangles
9 ā 12 Solve problems involving 45Ā°245Ā°290Ā° triangles
13 ā 16 Determine the area of 45Ā°245Ā°290Ā° triangles
17 ā 20 Solve problems involving 45Ā°245Ā°290Ā° triangles
21 ā 24 Construct 45Ā°245Ā°290Ā° triangles
5.5Properties of a 30Ā°ā60Ā°ā90Ā° Triangle
Vocabulary
1 ā 4 Determine the measure of indicated interior angles
5 ā 8Determine the length of the long leg and the hypotenuse of 30Ā°260Ā°290Ā° triangles
9 ā 12 Determine the lengths of the legs of 30Ā°260Ā°290Ā° triangles
13 ā 16Determine the length of the short leg and the hypotenuse of 30Ā°260Ā°290Ā° triangles
17 ā 20 Determine the area of 30Ā°260Ā°290Ā° triangles
21 ā 24 Construct 30Ā°260Ā°290Ā° triangles
Ā© Carnegie Learning
453A
ESSENTIAL IDEAS
Given the coordinates of two points, a third
point is located to form an equilateral
triangle, an isosceles triangle, a scalene
triangle, an acute triangle, an obtuse
triangle, and a right triangle.
Given the coordinates of three points,
algebra is used to describe characteristics
of the triangle.
TEXAS ESSENTIAL KNOWLEDGE
AND SKILLS FOR MATHEMATICS
(2) Coordinate and transformational geometry.
The student uses the process skills to understand
the connections between algebra and geometry
and uses the one- and two-dimensional
coordinate systems to verify geometric
conjectures. The student is expected to:
(B) derive and use the distance, slope, and
midpoint formulas to verify geometric
relationships, including congruence
of segments and parallelism or
perpendicularity of pairs of lines
5.1Name That Triangle!Classifying Triangles on the Coordinate Plane
LEARNING GOALS
In this lesson, you will:
Determine the coordinates of a third vertex of a triangle, given the coordinates
of two vertices and a description of the triangle.
Classify a triangle given the locations of its vertices on a coordinate plane.
Ā© Carnegie Learning
453B Chapter 5 Properties of Triangles
5
Overview
Students are given coordinates for 2 points and will determine a third set of coordinates that satisfy a
speci%c triangle description. Next, students are given the coordinates of three vertices of different
triangles and will describe the triangle using side lengths and angle measurements. Using a map,
students transfer a location onto a coordinate plane and answer questions related to the situation.
Ā© Carnegie Learning
5.1 Classifying Triangles on the Coordinate Plane 453C
5
Warm Up
The coordinates of two points A (26, 210) and B (4, 210) are given.
A (26, 210) B (4, 210)
C (26, y) C9 (4, y)
Describe all possible locations for the coordinates of point C such that triangle ABC is a right triangle.
Point C could have the coordinates (26, y) or (4, y), where y is any real number.
Ā© Carnegie Learning
5.1 Classifying Triangles on the Coordinate Plane 453
5
453
5.1
Because you may soon be behind the steering wheel of a car, it is important to
know the meaning of the many signs you will come across on the road. One of the
most basic is the yield sign. This sign indicates that a driver must prepare to stop to
give a driver on an adjacent road the right of way. The first yield sign was installed in
the United States in 1950 in Tulsa, Oklahoma, and was designed by a police officer of
the town. Originally, it was shaped like a keystone, but over time, it was changed.
Today, it is an equilateral triangle and is used just about everywhere in the world.
Although some countries may use different colors or wording (some countries call it a
āgive wayā sign), the signs are all the same in size and shape.
Why do you think road signs tend to be different, but basic, shapes, such as
rectangles, triangles, and circles? Would it matter if a stop sign was an irregular
heptagon? Does the shape of a sign make it any easier or harder to recognize?
LEARNING GOALS
In this lesson, you will:
Determine the coordinates of a third vertex of a triangle, given the coordinates
of two vertices and a description of the triangle.
Classify a triangle given the locations of its vertices on a coordinate plane.
Name That Triangle!Classifying Triangles on the Coordinate Plane
Students may need a
reminder of the three
classi%cations of triangles
by side, scalene,
isosceles, and equilateral,
and of the three
classi%cations by angle,
acute, obtuse, and right.
Help students create a
graphic organizer where
they write the type in
the %rst column, sketch
the type in the second
column, and description
in their own words in the
last column.
Ā© Carnegie Learning
454 Chapter 5 Properties of Triangles
5
Problem 1
Students are given two points
on the coordinate plane and will
determine all possible locations
for a third point that meet
speci%c triangular constraints
related to side lengths and
angle measures.
Grouping
Have students complete
Questions 1 through 3 with a
partner. Then have students
share their responses as
a class.
Guiding Questions for Share Phase, Questions 1 through 3
How can perpendicular
bisectors help determine
the possible locations for
point C?
How can the perpendicular
bisector of line segment AB
be helpful?
Is it possible for point C
to have an in%nite number
of locations to satisfy
this constraint?
How many possible locations
for point C are there if
triangle ABC is equiangular?
PROBLEM 1 Location, Location, Location!
1. The graph shows line segment AB with endpoints at A (26, 7) and B (26, 3).
Line segment AB is a radius for congruent circles A and B.
2 4 6x
28210 26 24 22
2
0
6
4
8
y
A
B
10
26
24
22
2. Using ___
AB as one side of a triangle, determine a
location for point C on circle A or on circle B such that
triangle ABC is:
a. a right triangle.
Point C can have an infinite number of locations
as long as the location satisfies one of the
following conditions:
Point C is located at the point (210, 7) or
(22, 7) on Circle A.
Point C is located at the point (210, 3) or (22, 3)
on Circle B.
b. an acute triangle.
Point C can have an infinite number of locations as
long as the location satisfies one of the following conditions:
Point C is located anywhere on circle A between the y-values of
3 and 7, except where x 5 26.
Point C is located anywhere on circle B between the y-values of
3 and 7, except where x 5 26.
If you are unsure
about where this point would lie, think about the steps it took to construct different triangles. Draw
additional lines or figures on your coordinate plane to
help you.
Ā© Carnegie Learning
5.1 Classifying Triangles on the Coordinate Plane 455
5
c. an obtuse triangle.
Point C can have an infinite number of locations as long as the location satisfies
one of the following conditions:
Point C is located at any point on circle A with a y-value greater than
7, except where x 5 26.
Point C is located at any point on circle B with a y-value less than
3, except where x 5 26.
3. Using ___
AB as one side of a triangle, determine the location for point C on circle A or on
circle B such that triangle ABC is:
a. an equilateral triangle.
Point C can have two possible locations. Circle A and circle B intersect at two
locations. Either point of intersection is a possible location for point C.
b. an isosceles triangle.
Point C can have an infinite number of locations as long as the location satisfies
one of the following conditions:
Point C is located anywhere on circle A, except where x 5 26.
Point C is located anywhere on circle B, except where x 5 26.
c. a scalene triangle.
Point C can have an infinite number of locations as long as the location is not at
any of the locations mentioned in parts (a) or (b).
Ā© Carnegie Learning
456 Chapter 5 Properties of Triangles
5
Problem 2
Students will graph three
points and use algebra to
determine the characteristics
of the triangle with respect to
the length of its sides and the
measures of the angles. They
use the Distance Formula to
classify the triangle as scalene,
isosceles, or equilateral. Next,
the slope formula and the
Pythagorean Theorem are used
to classify a triangle as a right
triangle. The second activity is
similar to the %rst activity.
Grouping
Have students complete
Questions 1 and 2 with a
partner. Then have students
share their responses as a class.
Guiding Questions for Share Phase, Questions 1 and 2
What formulas are used to
determine the length of the
sides of the triangle?
What formula helps to
determine if the triangle
contains a right angle?
What is the relationship
between the slopes of
perpendicular lines?
PROBLEM 2 Whatās Your Name Again?
1. Graph triangle ABC using points A (0, 24), B (0, 29), and C (22, 25).
28 26 24 22
22
24
26
20 4 6 8x
28
y
8
6
4
2
C
A
B
2. Classify triangle ABC.
a. Determine if triangle ABC is scalene, isosceles, or equilateral.
Explain your reasoning.
Because line segment AB is vertical, I can subtract the y-coordinates of the
endpoints to determine its length.
AB 5 24 2 (29)
5 5
BC 5 ā_______________________
(22 2 0)2 1 (25 2 (29))2
5 ā___________
(22)2 1 (4)2
5 ā_______
4 1 16
5 ā___
20
AC 5 ā_______________________
(22 2 0)2 1 (25 2 (24))2
5 ā_____________
(22)2 1 (21)2
5 ā______
4 1 1
5 ā__
5
Triangle ABC is scalene because no two side lengths
are equal.
These classifications are all about the
lengths of the sides. How can I determine the lengths of the sides of
this triangle?
Ā© Carnegie Learning
5.1 Classifying Triangles on the Coordinate Plane 457
5
?
b. Explain why triangle ABC is a right triangle.
Line segment AB is a vertical line on the y-axis. This means the slope is
undefined.
Slope of ___
AC :
m 5 y
2 2 y
1 _______ x2 2 x
1
m 5 25 2 (24)
__________
22 2 0 5 21 ___
22 5 1 __
2
Slope of ___
BC :
m 5 y
2 2 y
1 _______ x2 2 x
1
m 5 25 2 (29)
__________
22 2 0 5 4
___ 22
5 22
Triangle ABC is a right triangle. The slopes of the segments that form angle C are
negative reciprocals of each other, so they must be perpendicular, which means
they form a right angle.
c. Zach does not like using the slope formula. Instead, he decides to use the
Pythagorean Theorem to determine if triangle ABC is a right triangle because
he already determined the lengths of the sides. His work is shown.
Zach
a2 1 b2 5 c2
( ā__ 5 ) 2 1 ( ā
___ 20 ) 2 5 5 2
5 1 20 5 25
25 5 25
He determines that triangle ABC must be a right triangle because the sides satisfy
the Pythagorean Theorem. Is Zachās reasoning correct? Explain why or why not.
Yes. Zachās reasoning is correct. The Pythagorean Theorem only holds true for
right triangles. Because the side lengths of triangle ABC satisfy the
Pythagorean Theorem, Zach proved that triangle ABC is a right triangle.
Ā© Carnegie Learning
458 Chapter 5 Properties of Triangles
5
Grouping
Have students complete
Questions 3 and 4 with a
partner. Then have students
share their responses as a class.
Guiding Questions for Share Phase, Questions 3 and 4
What formulas are used to
determine the length of the
sides of the triangle?
What formula helps to
determine if triangle ABC
contains a right angle?
If triangle ABC is not a
right triangle, what are the
other possibilities?
3. Graph triangle ABC using points A (22, 4), B (8, 4), and C (6, 22).
28 26 24 22
22
24
26
20 4 6 8x
28
y
8
6
4
2
C
A B
4. Classify triangle ABC.
a. Determine if triangle ABC is a scalene, an isosceles, or an equilateral triangle.
Explain your reasoning.
Line segment AB is horizontal so I can determine its length by subtracting the
x-coordinates of its endpoints.
AB 5 8 2 (22)
5 10
BC 5 ā___________________
(6 2 8)2 1 (22 2 4)2
5 ā_____________
(22)2 1 (26)2
5 ā_______
4 1 36
5 ā___
40
AC 5 ā_____________________
(6 2 (22))2 1 (22 2 4)2
5 ā___________
(8)2 1 (26)2
5 ā________
64 1 36
5 ā____
100
5 10
Because sides AB and AC are equal, triangle ABC must be isosceles. The triangle
is not equilateral, though, because the length of the third side, BC, is not equal to
the other two lengths.
Ā© Carnegie Learning
5.1 Classifying Triangles on the Coordinate Plane 459
5
b. Determine if triangle ABC is a right triangle. Explain your reasoning. If it is not a right
triangle, use a protractor to determine what type of triangle it is.
Line segment AB is a horizontal line, so the slope is 0.
Slope of line segment BC:
m 5 y
2 2 y
1 _______ x2 2 x
1
m 5 4 2 (22)
________ 8 2 6
5 6 __ 2 5 3
Slope of line segment AC:
m 5 y
2 2 y
1 _______ x2 2 x
1
m 5 4 2 (22)
________ 22 2 6
5 6 ___ 28
5 2 3
__ 4
Triangle ABC is not a right triangle because none of the line segments has a
perpendicular relationship with another line segment.
/A 5 30Ā°
/B 5 70Ā°
/C 5 80Ā°
Because all three measures have measures that are less than 90Ā°, triangle ABC must
be an acute triangle.
Ā© Carnegie Learning
460 Chapter 5 Properties of Triangles
5
Problem 3
Students use a map to
determine approximate
coordinates of three
destinations. They connect the
locations to form a triangle and
classify the triangle.
Grouping
Have students complete
Question 1 with a partner.
Then have students share their
responses as a class.
Guiding Questions for Share Phase, Question 1
How is using the origin as
a location for one of the
cities helpful?
Which city did you graph
%rst? Why?
Can you locate a second city
on the x-axis? Which city?
Is the third city located above
or below the x-axis? Why?
How did you determine the
location of the third city?
PROBLEM 3 Indiaās Golden Triangle
1. Indiaās Golden Triangle is a very popular tourist destination. The vertices of the triangle
are the three historical cities of Delhi, Agra (Taj Mahal), and Jaipur.
The locations of these three cities can be represented on the coordinate plane
as shown.
x
y
(0, 0) (134, 0)
(100, 105)
Rohtak
Alwar
Fatehpur Sikri
Delhi
Jaipur Agra
Classify Indiaās Golden Triangle.
JD 5 ā_____________________
(100 2 0)2 1 (105 2 0)2 DA 5 ā________________________
(100 2 134)2 1 (105 2 0)2
5 ā___________
1002 1 1052 5 ā_____________
(234)2 1 1052
5 ā________________
10,000 1 11,025 5 ā______________
1156 1 11,025
5 ā_______
21,025 5 ā_______
12,181
5 145 ĀÆ 110.37
Line segment JA is horizontal so I can determine its length by subtracting the
x-coordinates of its endpoints.
JA 5 134 2 0
5 134
Indiaās Golden Triangle is an acute scalene triangle because each side is a different
length and each angle is less than 90 degrees.
Be prepared to share your solutions and methods.