Lesson 8-R

Chapter 8 Review

5-Minute Check on Chapter 85-Minute Check on Chapter 85-Minute Check on Chapter 85-Minute Check on Chapter 8 Transparency 9-1

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

Complete each statement about parallelogram LMNO

1. LM _______

2. MN _______

3. OLM _______

4. MP _______

5. Find the measure of each interior angle

6. What is the measure of each interior angle of a regular pentagon?Standardized Test Practice:

A CB D90° 108°

L

O N

C

M

135°120°

P

(4y + 5)°

(8y - 5)° (4y + 5)°

(8y - 5)°

D

A B

ON

LO

ONM

PO

A = C = 65°

B

B = D = 115°

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 angleInterior 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°.

Sides NameSum of

Interior ’sOne

Interior One

Exterior

3 180 60

5 72

Heptagon 900 128.57

Polygon HierarchyPolygons

Squares

RhombiRectangles

Parallelograms Kites Trapezoids

IsoscelesTrapezoids

Quadrilaterals

Sides Name

3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon

12 Dodecagon

n N-gon

Polygon Venn Diagram

Quadrilaterals

Parallelograms

Rectangles

IsoscelesTrapezoids

Trapezoids

Rhombi

SquaresKites

Quadrilateral Characteristics SummaryConvex Quadrilaterals

Squares

RhombiRectangles

Parallelograms Trapezoids

IsoscelesTrapezoids

Opposite sides parallel and congruentOpposite angles congruentConsecutive angles supplementaryDiagonals bisect each other

Bases ParallelLegs are not ParallelLeg angles are supplementary Median is parallel to basesMedian = ½ (base + base)

Angles all 90°Diagonals congruent

Diagonals divide into 4 congruent triangles

All sides congruentDiagonals perpendicularDiagonals bisect opposite angles

Legs are congruent Base angle pairs congruent Diagonals are congruent

4 sided polygon4 interior angles sum to 3604 exterior angles sum to 360

In the isosceles trapezoidEF is a median,

2x + 8

6x - 6

25 3y - 6

y + 4 9z°

6z°A B

C D

E F

In the rhombus,

J K

L M

N

In the square,

W P

BH

A35°

35

3x + 8

m° 2y -1 25

R S

TU

V

2k°

3x - 8

16

4y + 4

In the rectangle,

m°3x+5

353y6x

P Q

R S

In the parallelogram,

9z 18

3z

2y

4x

24

z

t

54°

2z + 6

5t°8t°

2t°3t°

Example Problems 2

24

w°

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°.

Sides NameSum of

Interior ’sOne

Interior One

Exterior

3 Triangle 180 60 120

5 Pentagon 540 108 72

7 Heptagon 900 128.57 51.43

S = (n – 2) 180 = (16 – 2) 180 = 14 180 = 2520

S = 360

Int + Ext = 180 so Ext = 40

n = 360 / Ext = 360 / 40 = 9

In the rhombus,

J K

L M

N

In the square,

W P

BH

A35°

35

3x + 8

m° 2y -1 25

R S

TU

V

2k°

3x - 8

16

4y + 4

In the rectangle,

9z 18

3z

2y

4x

24

z

t

54°

Example Solutions 2

diagonals =and bisected

25 = 2y – 126 = 2y13 = 3

Opposite sides =

35 = 3x + 827 = 3x9 = x

diagonals bisected

z = t8 = t

all sides =

3z = 4x = 2y = 24z = 8, x = 6, y = 12

all sides =

4y + 4 = 16 = 3x – 84y = 12 24 = 3x y = 3 8 = x

diagonals

2k = 90k = 45

2 pairs isosceles ∆

35 + 35 + x = 180 x + m = 180 (L pr) m = 70

diagonals bisect anglesw = 54

w°

In the isosceles trapezoidEF is a median,

2x + 8

6x - 6

25 3y - 6

y + 4 9z°

6z°A B

C D

E F

m°3x+5

353y6x

P Q

R S

In the parallelogram,

2z + 6

5t°8t°

2t°3t°

Example Solutions 2 Cont

Consecutive ’s supplementary

8t + 5t + 2t + 3t = 180 18t = 180 t = 10

diagonals bisected

35 = 3x + 530 = 3x10 = x

diagonals bisected

3y = 6x 3y = 60y = 20

24

opposite sides =

24 = 2z + 618 = 2z9 = z

isosceles legs =

y + 4 = 3y – 6 10 = 2y 5 = yisosceles leg ’s

supplementary

6z + 9z = 180 15z = 180 z = 12

isosceles base ’s =

6z = m72 = m

median = ½(sum of bases) 25 = ½(6x – 6 + 2x + 8) 50 = 6x – 6 + 2x + 8 50 = 8x + 2 48 = 8x 6 = x

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