1. Lesson 8-R
Chapter 8 Review

2. 5-Minute Check on Chapter 8
Transparency 9-1
5-Minute Check on Chapter 8
Complete each statement about parallelogram LMNO
LM  _______
MN  _______
OLM  _______
MP  _______
Find the measure of each interior angle
What is the measure of each interior angle of a regular pentagon? ON LO
ONM PO
A = C = 65° B
B = D = 115° L M P O N A B
(4y + 5)°
(8y - 5)°
(8y - 5)°
(4y + 5)° D C
Standardized Test Practice: A
90° B
108° C
120° D
135°
Click the mouse button or press the Space Bar to display the answers.

3. Angles in Convex Polygons
Interior angle + exterior angle = 180°
They are a Linear Pair
Sum of Interior angles, S = (n-2)  180°
One Interior angle = S / n = (n-2)  180°/n
Sum of Exterior angles = 360°
Number of sides, n = 360° / Exterior angle
Exterior angle
Interior angle

4. Example Problems 1 Find the sum of the interior angles in a 16-gon
Sides
Name
Sum of Interior ’s
One Interior 
One Exterior  3
180 60 5 72
Heptagon
900
128.57
Find the sum of the interior angles in a 16-gon
Find the sum of the exterior angles in a 16-gon
Find the number of sides of a polygon if an interior angle is 140°.

5. Polygon Hierarchy Polygons Quadrilaterals Parallelograms Kites
Sides
Name 3
Triangle 4
Quadrilateral 5
Pentagon 6
Hexagon 7
Heptagon 8
Octagon 9
Nonagon 10
Decagon 12
Dodecagon n
N-gon
Quadrilaterals
Parallelograms
Kites
Trapezoids
Isosceles Trapezoids
Rectangles
Rhombi
Squares

6. Polygon Venn Diagram Quadrilaterals Trapezoids Parallelograms
Rectangles
Isosceles Trapezoids
Trapezoids
Rhombi
Squares
Kites

7. Quadrilateral Characteristics Summary
Convex Quadrilaterals
4 sided polygon
4 interior angles sum to 360
4 exterior angles sum to 360
Parallelograms
Trapezoids
Bases Parallel
Legs are not Parallel
Leg angles are supplementary
Median is parallel to bases Median = ½ (base + base)
Opposite sides parallel and congruent
Opposite angles congruent
Consecutive angles supplementary
Diagonals bisect each other
Rectangles
Rhombi
Isosceles Trapezoids
All sides congruent
Diagonals perpendicular
Diagonals bisect opposite angles
Angles all 90°
Diagonals congruent
Legs are congruent
Base angle pairs congruent
Diagonals are congruent
Squares
Diagonals divide into 4 congruent triangles

8. Example Problems 2 W P B H A R S T U V 9z 18 24 J K L M N t 3z z 4x w°
In the rectangle,
In the square, W P B H A
35° 35
3x + 8 m°
2y -1 25 R S T U V
2k°
3x - 8 16
4y + 4 9z 18
In the rhombus, 24 J K L M N t 3z z 4x w°
54°
In the isosceles trapezoid
EF is a median, 2y
In the parallelogram, P Q
6x - 6
3x+5 6x A B
6z° m° 24
3y - 6
2z + 6 3y 35 25 E F
8t°
3t°
y + 4
5t°
2t°
9z° R S C
2x + 8 D

9. Example Solutions 1 Find the sum of the interior angles in a 16-gon
Sides
Name
Sum of Interior ’s
One Interior 
One Exterior  3
Triangle
180 60
120 5
Pentagon
540
108 72 7
Heptagon
900
128.57
51.43
Find the sum of the interior angles in a 16-gon
Find the sum of the exterior angles in a 16-gon
Find the number of sides of a polygon if an interior angle is 140°.
S = (n – 2)  180 = (16 – 2)  180 = 14  180 = 2520
S = 360
Int  + Ext  = so Ext  = 40
n = 360 / Ext  = 360 / 40 = 9

10. Example Solutions 2 2 pairs isosceles ∆ 35 + 35 + x = 180
x + m = 180 (L pr)
m = 70
In the rectangle,
In the square, W P B H A
35° 35
3x + 8 m°
2y -1 25 R S T U V
2k°
3x - 8 16
4y + 4 9z 18
In the rhombus,
Opposite sides =
35 = 3x + 8
27 = 3x
9 = x 24 J K L M N
all sides =
4y + 4 = 16 = 3x – 8
4y = = 3x
y = = x t 3z z 4x w°
54°
diagonals =
and bisected
25 = 2y – 1
26 = 2y
13 = 3 2y
all sides =
3z = 4x = 2y = 24
z = 8, x = 6, y = 12
diagonals bisected
z = t
8 = t
diagonals 
2k = 90
k = 45
diagonals bisect angles
w = 54

11. Example Solutions 2 Cont
isosceles legs =
y + 4 = 3y – 6
10 = 2y
5 = y
diagonals bisected
35 = 3x + 5
30 = 3x
10 = x
opposite sides =
24 = 2z + 6
18 = 2z
9 = z
isosceles leg ’s
supplementary
6z + 9z = 180
15z = 180
z = 12
diagonals bisected
3y = 6x
3y = 60
y = 20
Consecutive ’s
supplementary
8t + 5t + 2t + 3t = 180
18t = 180
t = 10
isosceles
base ’s =
6z = m
72 = m
In the isosceles trapezoid
EF is a median,
In the parallelogram,
median = ½(sum of bases)
25 = ½(6x – 6 + 2x + 8)
50 = 6x – 6 + 2x + 8
50 = 8x + 2
48 = 8x
6 = x P Q
6x - 6
3x+5 6x A B
6z° m° 24
3y - 6
2z + 6 3y 35 25 E F
8t°
3t°
y + 4
5t°
2t°
9z° R S C
2x + 8 D

12. Do you know your characteristics?
Homework assignment
Chapter 8 Review Problems

13. Summary & Homework Summary: Homework:
Interior and Exterior angles make a linear pair (=180)
Sum of interior angles = (n - 2)  180
Sum of exterior angles = 360 (no matter the size)
Number of sides = 360 / exterior angle
Quadrilateral characteristics are very important for solving problems and verifying figures
Reminder: Sum of triangle angles = 180
Medians in trapezoids are similar to mid-segments
Homework:
study for the test