1. Lesson 7-R
Chapter 7 Review

2. 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

3. 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°.

4. 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

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

6. 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
Kites
2 congruent sides (consecutive)
Diagonals perpendicular
Diagonals bisect opposite angles
One diagonal bisected
One pair of opposite angle congruent
Rectangles
Rhombi
Isosceles Trapezoids
Angles all 90°
Diagonals congruent
All sides congruent
Diagonals perpendicular
Diagonals bisect opposite angles
Squares
Legs are congruent
Base angle pairs congruent
Diagonals are congruent
Diagonals divide into 4 congruent triangles

7. 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

8. 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

9. 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

10. 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

11. 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