final exam key concepts. vertical angles find the value of “x”. 68 = 2x + 32 36 = 2x 18 = x
TRANSCRIPT
Final Exam
Key Concepts
Vertical Angles
• Find the value of “x”.
(2x + 32)68
68 = 2x + 32
36 = 2x
18 = x
Segments and Lengths• In the diagram, and S is the
midpoint of . QR = 4, and ST = 5. Find the following values.
1. RS =
2. PR =
3. PQ =
PR RTRT
5
P Q R S T10
6
Parallel Lines
• Find the missing values.
z
y
x
110
70o
70o
110o
Straight Line is 180o
• Find the difference between the larger angle and the smaller angle.
(4x + 19)
(3x)3x + (4x + 19) = 180
7x + 19 = 180
7x =161
x =23
3(23) = 69 and 4(23) + 19 = 111
111 – 69 = 42
Angles of Triangle = 180o
• The angles of a triangle are in the ratio of 1:2:3. Find the measure of each angle.
1x + 2x + 3x = 180
6x = 180
x = 30
30o, 60o, 90o
Exterior Angle Theorem
• Find the missing value.
x
Note: Figure not drawn to scale.
35
25
35o + 25o = 60o
60o
Angles and Sides of a Triangle
• Remember, largest angle is OPPOSITE of longest side….
• And smallest angle is OPPOSITE of smallest side.
45
67
68
B
C
A
BC
AB
TRIANGLE INEQUALITY THEOREM
• The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
• Is it possible for a triangle to have the following lengths?
3, 6, 8 6 + 3 = 9 > 8 YES
Three Theorems about Triangles
1) If c2 = a2 + b2, then the triangle is a right triangle.
2) If c2 < a2 + b2, then the triangle is an acute triangle.
3) If c2 > a2 + b2, then the triangle is an obtuse triangle.
12
RIGHT TRIANGLES
Special Right Triangles
Pythagorean Theorem
62 + 72 = x2
36 + 49 = x2
85 = x2
85x
Pythagorean Triples
3, 4, 5
5, 12, 13
8, 15, 17
7, 24, 25
Quadrilaterals
Quadrilaterals
• Complete the worksheet.
5 Ways to Prove Parallelograms
1. Both pairs of opposite sides are parallel
2. Both pairs of opposite sides are congruent
3. Both pairs of opposite angles are congruent
4. One pair of opposite sides congruent and parallel
5. Diagonals bisect each other.
INTERIOR MEASURES
SUM of the INTERIOR
Measures of Any Polygon
(n – 2)180o
(4-2)180o = 360o (8-2)180o = 1080o
EXTERIOR ANGLES
Sum of the measures of the EXTERIOR angles of any polygon
= 360o
e s
r
o
h
p
a
r
c
h + o + r + s + e = 360oc + r + a + p = 360o
REGULAR POLYGON
A polygon that is both equilateral and equiangular.
Areas
2
1 2
1 2
AREA
Rectangles =
Squares =
Parallelogram
1Triangles =
21
Rhombus= 2
1Regular Polygon =
21
Trapezoids =2
bh
s
bh
bh
d d
ap
h b b
Circles
2
2
CIRCLES
Area of Circle =
Circumference of Circle = 2
Arc length = 2360
Area of a sector = 360
r
r
xr
xr
Angles in a Circle
central angles =
1inscribed angles =
21
interior angles = ( )21
exterior angles = ( )2
arc
arc
big small
big small
80o
50o
50o
30o
Lengths in a Circle
12(9) = 18x
108 = 18x
6 = x
Lengths in a Circle
3(8) = 2(12)
24 = 24
Lengths in a Circle
122 = x(x+12+x)
144 = 2x2 + 12x
x = 8
SAT Formulas