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Lesson 8-R. Chapter 8 Review. 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

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Lesson 8 r

Lesson 8-R

Chapter 8 Review


Lesson 8 r

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

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.


Angles in convex polygons

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

Interior angle

Exterior angle


Example problems 1
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
Polygon Hierarchy

Polygons

Quadrilaterals

Parallelograms

Kites

Trapezoids

IsoscelesTrapezoids

Rectangles

Rhombi

Squares


Polygon venn diagram
Polygon Venn Diagram

Quadrilaterals

Trapezoids

IsoscelesTrapezoids

Parallelograms

Rhombi

Kites

Rectangles

Squares


Quadrilateral characteristics summary
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


Example problems 2

3x + 8

35°

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 2

In the square,

In the rectangle,

18

9z

In the rhombus,

24

t

3z

4x

z

54°

In the isosceles trapezoid

EF is a median,

2y

In the parallelogram,

P

Q

6x - 6

A

B

3x+5

6x

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
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


Example solutions 2

3x + 8

35°

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 2

2 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

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
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°

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
Do you know your characteristics?

  • Homework assignment

  • Chapter 8 Review Problems


Summary homework
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