8.1 Find Angle Measures in Polygons - PAP …papgeometry.weebly.com/uploads/2/2/8/6/22869852/8.1_oh_rearranged.pdf8.1 Find Angle Measures in Polygons ... •What if the sum of the

8.1 Find Angle Measures in

Polygons

Goals -find the sum of measures of

interior and exterior angles

-how to use polygons to solve

real life problems.

Polygon – Greek – many sides • 3 or more sides

• lines

• Each side must intersect exactly 2 other

sides – once at each end point.

• No two sides with a common endpoint are

collinear.

Vertices identified by letters

(A, B, C, D, E …)

• TO NAME POLYGONS: list vertices

consecutively ABCDEF, CDEFAB, etc.

• Diagonal – a segment that joins two

nonconsecutive vertices (BF, FC, FD)

B C

D

E F

A.

CONVEX

No line contains a side of the polygon in the

interior of the polygon

concave

Polygons can be class by number

of sides

Triangles (3) Heptagon (7)

Quadrilateral (4) Octagon (8)

Pentagon (5) Nonagon (9)

Hexagon (6) Decagon (10)

n-gon (n sides)

A Polygon is:

• Equilateral if all sides are congruent.

• Equiangular if all its angles are congruent.

• Regular if it is equilateral and equiangular.

Discover – Interior Angles 1. Divide each polygon into triangles, by drawing all the

diagonals from one vertex.

2. Find the measure of the interior angles of each polygon. Polygon # of # of # of interior sides vertices triangles measure Triangle 3 3 1 1 * 180

-

Will your formula work on all polygons?

Will the formula (n-2) 180 work as well?

The sum of the measures of the interior <‘s

of a convex n-gon is:

(n-2)(180º)

The measure of each internal angle of a

regular n-gon:

(n-2)(180º)

n

For a regular pentagon

Sum of the interior angles =

(5-2)(180) = 540º

Each angle = 540

1085

Number of

angles or sides

8x + 1

x - 3

2x + 32

5x - 4 3x + 1

x- 7 8x

2x + 9

4x

2x - 1

2x + 3

7x + 5

9x - 1

5x + 4

x + 1

4x 3x + 19

x - 3

2x + 1

2x 3x + 10

9x + 1

x + 2

1. 2.

3. 4.

281° 32°

102°

171 106

28° 176°

53°

88°

43°

47°

159°

197°

114°

23°

168°

145°

39°

85°

84°

136°

379°

44°

1. 2.

3. 4.

X = 35

X = 42

X = 22

X = 22

• DISCOVER EXTERIOR ANGLES

• Step 1:

• Find the sum of the measures of the exterior angles of each polygon.

• Step 2:

• Make a conjecture about the sum of the measures of the exterior angles of any polygon:

• Step 3:

• Find the measure of a single exterior angle for

•

• a triangle ______

•

• a quadrilateral _____

•

• an octagon ____

•

• an n-gon ________

Exterior angles:

The sum of the measures of the exterior

angles is 360º

The measure of each exterior angle

Of an n-gon is

360º

n

You try!

• What is the sum of the

exterior angles of a

pentagon?

• Given a regular

pentagon, what does

each exterior angles

measure?

What about non-regular polygons?

7

2

1 6

9

10

8 3

4

5

Finding measures

Given m 2 = 100º, m 8 = 40º, m 4 = m 5 = 110º.

Find the measure of all the other angles above.

100

80 100 110

70

110

70

40

140

80

Example: Exterior Angle Sum

Theorem

SOLUTION:

You try!

• What if the sum of the interior angles of a polygon equal 7020 ; How many sides does it have?

• What if the measure of an interior angle of a regular polygon measures 108 . How many sides does it have?

41 sides

5 sides

Rally table – regular polygons

• Handout on back of discovery ws

1. 5

2. 20

3. 90

4. 30

Find My Sum Activity

• Person 1: Select a quadrant 1 - 4.

• Person 2: Calculates the sum of the angles of the polygon requested, explaining to the team each step he/she does. Record the calculations on the team paper and initial the work

• Person 3: Solves for x, explaining to the team each step they do. Record the calculations and initial the work.

• Person 4: Check the work. Coach if necessary and/or praise student 2 & 3.

• The next person in a clockwise direction selects the next polygon to do and steps 2-4 begin again, rotating the roles to the next student. Repeat until all four polygons are done.

Group work

• Everybody does their own sheet,

collaborate, agree and check answers

• I will randomly select and grade one sheet

for everybody in the group.

Homework ws 8.1