3) Solve for x..
30 
(4x + 2)°
(8 + 6x) 
1

5) Find the values of the variables and then the measures of the angles .
z° w° y° x°
30°
(2y – 6)°
2

Geometry
3

Q
VERTEX
R
SIDE
P


VERTEX
S


T
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.





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/
D
A
E
B
Figures A, B, and C are polygons.
F
C

Polygons are named by the number of sides they have –
MEMORIZE
Number of sides
3
Type of Polygon
Triangle
4
5
Quadrilateral
Pentagon
6
7
8
9
10
12 n
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Dodecagon n-gon

A convex polygon has no diagonal with points outside the polygon.
A concave polygon has at least one diagonal with points outside the polygon
7

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.
8

Measures of Interior and Exterior
Angles

For instance . . . Complete this table
Polygon
Triangle
Quadrilateral
# of sides
3
# of triangles
1
Sum of measures of interior  ’s
1●180  =180 
2●180  =360 
Pentagon
Hexagon
Nonagon (9) n-gon n
9

6) Find the sum of the interior angles of a dodecagon.
10

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.
11

Ex. 1: Finding measures of Interior
Angles of Polygons

Find the value of x in the diagram shown:
88 
142 
136 
136 
105  x 
12



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 
13

136  + 136  + 88  +
142  + 105  +x  =
720  .
607 + x = 720
X = 113
•The measure of the sixth interior angle of the hexagon is 113  .
14


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:
1 or
● (n-2) ● 180  n
( n

2 )( 180 ) n
15

n=10

( n

2)(180) n

10

8(180)
10

144

144
16

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
17

( n

2 )( 180 ) n
= 140 
Corollary to Thm. 11.1
(n – 2) ●180  = 140  n
Multiply each side by n.
180n – 360 = 140  n
40n = 360 n = 90
Distributive Property
Addition/subtraction props.
Divide each side by 40.
18

19

3-10
3-10
20

Ex. 3: Finding the Measure of an
Exterior Angle
21

Ex. 3: Finding the Measure of an
Exterior Angle
Simplify.
3-10
22

Ex. 3: Finding the Measure of an
Exterior Angle
3-10
23

Using Angle Measures in Real Life
Ex. 4: Finding Angle measures of a polygon
24

Using Angle Measures in Real Life
Ex. 5: Using Angle Measures of a Regular
Polygon
25

Using Angle Measures in Real Life
Ex. 5: Using Angle Measures of a Regular
Polygon
26

Using Angle Measures in Real Life
Ex. 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.
b.
135 ° ?
145 ° ?
27

Using Angle Measures in Real Life
Ex. : Finding Angle measures of a polygon
28