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### Lesson 8-R

### Angles in Convex Polygons

Chapter 8 Review

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

L

M

LO

P

ONM

O

N

PO

A

B

(8y - 5)°

(4y + 5)°

A = C = 65°

B = D = 115°

(4y + 5)°

(8y - 5)°

D

C

Standardized Test Practice:

A

90°

B

108°

C

120°

D

135°

B

Click the mouse button or press the Space Bar to display the answers.

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

Interior angle

Exterior angle

Example Problems 1

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°.

Polygon Hierarchy

Polygons

Quadrilaterals

Parallelograms

Kites

Trapezoids

IsoscelesTrapezoids

Rectangles

Rhombi

Squares

Polygon Venn Diagram

Quadrilaterals

Trapezoids

IsoscelesTrapezoids

Parallelograms

Rhombi

Kites

Rectangles

Squares

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 basesMedian = ½ (base + base)

Opposite sides parallel and congruent

Opposite angles congruent

Consecutive angles supplementary

Diagonals bisect each other

Rectangles

Rhombi

IsoscelesTrapezoids

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

35°

m°

2y -1

25

35

3x - 8

R

S

J

K

V

16

2k°

N

L

M

U

T

4y + 4

W

P

A

B

H

Example Problems 2In the square,

In the rectangle,

18

9z

In the rhombus,

24

t

3z

4x

z

w°

54°

In the isosceles trapezoid

EF is a median,

2y

In the parallelogram,

P

Q

6x - 6

A

B

3x+5

6x

m°

6z°

24

3y - 6

25

3y

2z + 6

35

E

F

3t°

8t°

y + 4

9z°

5t°

2t°

S

R

C

D

2x + 8

Example Solutions 1

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 = 180 so Ext = 40

n = 360 / Ext = 360 / 40 = 9

35°

m°

2y -1

25

35

3x - 8

R

S

J

K

V

16

2k°

N

L

M

U

T

4y + 4

W

P

A

B

H

Example Solutions 22 pairs isosceles ∆

35 + 35 + x = 180

x + m = 180 (L pr)

m = 70

In the square,

In the rectangle,

18

9z

In the rhombus,

Opposite sides =

35 = 3x + 8

27 = 3x

9 = x

24

all sides =

4y + 4 = 16 = 3x – 8

4y = 12 24 = 3x

y = 3 8 = x

t

3z

4x

z

w°

54°

2y

diagonals =

and bisected

25 = 2y – 1

26 = 2y

13 = 3

diagonals bisected

z = t

8 = t

all sides =

3z = 4x = 2y = 24

z = 8, x = 6, y = 12

diagonals

2k = 90

k = 45

diagonals bisect angles

w = 54

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

isosceles

base ’s =

6z = m

72 = m

Consecutive ’s

supplementary

8t + 5t + 2t + 3t = 180

18t = 180

t = 10

In the isoscelestrapezoid

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

A

B

3x+5

6x

6z°

m°

24

3y - 6

25

3y

2z + 6

35

E

F

3t°

8t°

y + 4

9z°

5t°

2t°

S

R

C

D

2x + 8

Do you know your characteristics?

- Homework assignment
- Chapter 8 Review Problems

Summary & Homework

- Summary:
- 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

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