Pre-AP Bellwork 10-193) Solve for x..

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30

(4x + 2)°

(8 + 6x)

Pre-AP Bellwork 10-24

5) Find the values of the variables and then the measures of the angles.

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z°

w°

x°

y°

30°

(2y – 6)°

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3-4 Polygon Angle-Sum Theorem

Geometry

Definitions:

Polygon—a plane figure that meets the following conditions: It is formed by 3 or more segments called

sides, such that no two sides with a common endpoint are collinear.

Each side intersects exactly two other sides, one at each endpoint.

Vertex – each endpoint of a side. Plural is vertices. You can name a polygon by listing its vertices consecutively. For instance, PQRST and QPTSR are two correct names for the polygon above.

VERTEX

VERTEX

Q

R

ST

PSIDE

Example 1: Identifying Polygons

State whether the figure is a polygon. If it is not, explain why.

Not D – has a side that isn’t a segment – it’s an arc.

Not E– because two of the sides intersect only one other side.

Not F because some of its sides intersect more than two sides/

F

E

D

CBA

Figures A, B, and C are polygons.

Polygons are named by the number of sides they have – MEMORIZE

Number of sides Type of Polygon

3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon

12 Dodecagon

n n-gon

Convex or Concave???

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A convex polygon has no diagonal with points outside the polygon.

A concave polygon has at least one diagonal with points outside the polygon

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Measures of Interior and Exterior Angles

You have already learned the name of a polygon depends on the number of sides in the polygon: triangle, quadrilateral, pentagon, hexagon, and so forth. The sum of the measures of the interior angles of a polygon also depends on the number of sides.

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Measures of Interior and Exterior Angles

For instance . . . Complete this table

Polygon # of sides

# of triangles

Sum of measures of interior ’s

Triangle 3 1 1●180=180Quadrilateral 2●180=360

Pentagon

Hexagon

Nonagon (9)

n-gon n

Pre-AP Bellwork 10 - 24 6) Find the sum of the interior angles of

a dodecagon.

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11

Measures of Interior and Exterior Angles

What is the pattern? (n – 2) ● 180.

This relationship can be used to find the measure of each interior angle in a regular n-gon because the angles are all congruent.

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Ex. 1: Finding measures of Interior Angles of Polygons

Find the value of x in the diagram shown:

88

136

136

142

105

x

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SOLUTION:

S(hexagon)= (6 – 2) ● 180 = 4 ● 180 = 720.

Add the measure of each of the interior angles of the hexagon.

88

136

136

142

105

x

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SOLUTION:

136 + 136 + 88 + 142 + 105 +x = 720.

607 + x = 720

X = 113

•The measure of the sixth interior angle of the hexagon is 113.

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Polygon Interior Angles Theorem

The sum of the measures of the interior angles of a convex n-gon is (n – 2) ● 180

COROLLARY:The measure of

each interior angle of a regular n-gon is:

n

1● (n-2) ● 180

n

n )180)(2( or

EX.2 Find the measure of an interior angle of a decagon….

n=10

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( 2)(180)n

n

(10 2)(180)

10

8(180)

10

144

144

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Ex. 2: Finding the Number of Sides of a Polygon

The measure of each interior angle is 140. How many sides does the polygon have?

USE THE COROLLARY

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Solution:

n

n )180)(2( = 140

(n – 2) ●180= 140n

180n – 360 = 140n

40n = 360

n = 90

Corollary to Thm. 11.1

Multiply each side by n.

Distributive Property

Addition/subtraction props.

Divide each side by 40.

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Copy the item below.

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EXTERIOR ANGLE THEOREMS

3-10

3-10

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Ex. 3: Finding the Measure of an Exterior Angle

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Ex. 3: Finding the Measure of an Exterior Angle

3-10

Simplify.

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Ex. 3: Finding the Measure of an Exterior Angle

3-10

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Using Angle Measures in Real LifeEx. 4: Finding Angle measures of a polygon

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Using Angle Measures in Real LifeEx. 5: Using Angle Measures of a Regular Polygon

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Using Angle Measures in Real LifeEx. 5: Using Angle Measures of a Regular Polygon

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Using Angle Measures in Real LifeEx. 5: Using Angle Measures of a Regular Polygon

Sports Equipment: If you were designing the home plate marker for some new type of ball game, would it be possible to make a home plate marker that is a regular polygon with each interior angle having a measure of:

a. 135°?b. 145°?

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Using Angle Measures in Real LifeEx. : Finding Angle measures of a polygon