1. Lesson 8-1
Angles of Polygons

2. Objectives
Find the sum of the measures of the interior angles of a polygon
Sum of Interior angles = (n-2) • 180
One Interior angle = (n-2) • 180 / n
Find the sum of the measures of the exterior angles of a polygon
Sum of Exterior angles = 360
One Exterior angle = 360/n
Exterior angle + Interior angle = 180

3. Vocabulary
Diagonal – a segment that connects any two nonconsecutive vertices in a polygon.

4. Angles in a Polygon 1 2 3 4 5 6 7 8 Octagon n = 8
8 180° - 360° (center angles)
= (8-2) • 180 = 1080
Sum of Interior angles = (n-2) • 180

5. Sum of Interior Angles: so each interior angle is (n – 2) * 180
Angles in a Polygon
Sum of Interior Angles:
(n – 2) * where n is number of sides
so each interior angle is (n – 2) * 180 n
Octagon
n = 8
Interior Angle
Sum of Exterior Angles:
so each exterior angle is n
Interior Angle + Exterior Angle = 180
Exterior Angle
Octagon
Sum of Exterior Angles: 360
Sum of Interior Angles: 1080
One Interior Angle: 135
One Exterior Angle: 45

6. Polygons Sides Name Sum of Interior Angles One Interior Angle
Sum Of Exterior Angles
One Exterior Angles 3
Triangle
180 60
360
120 4
Quadrilateral 90 5
Pentagon
540
108 72 6
Hexagon
720 7
Heptagon
900
129 51 8
Octagon
1080
135 45 9
Nonagon
1260
140 40 10
Decagon
1440
144 36 12
Dodecagon
1800
150 30 n
N - gon
(n-2) * 180
180 – Ext 
360 ∕ n =

7. Interior Angle Sum Theorem
ARCHITECTURE A mall is designed so that five walkways meet at a food court that is in the shape of a regular pentagon. Find the sum of measures of the interior angles of the pentagon.
Since a pentagon is a convex polygon, we can use the Angle Sum Theorem.
Interior Angle Sum Theorem
Simplify.
Answer: The sum of the measures of the angles is 540.
Example 1-1a

8. Interior Angle Sum Theorem
The measure of an interior angle of a regular polygon is 135. Find the number of sides in the polygon.
Use the Interior Angle Sum Theorem to write an equation to solve for n, the number of sides.
Interior Angle Sum Theorem
Distributive Property
Subtract 135n from each side.
Add 360 to each side.
Divide each side by 45.
Answer: The polygon has 8 sides.
Example 1-2a

9. SHORT CUT!!
The measure of an interior angle of a regular polygon is 135. Find the number of sides in the polygon.
Exterior angle = 180 – Interior angle = 45
n = = = 8
Ext 

10. Find the measure of each interior angle.
Example 1-3a

11. At each vertex, extend a side to form one exterior angle.
Find the measures of an exterior angle and an interior angle of convex regular nonagon ABCDEFGHJ.
At each vertex, extend a side to form one exterior angle.
Answer: Measure of each exterior angle is 40. Since each exterior angle and its corresponding interior angle form a linear pair, the measure of the interior angle is 180 – 40 or 140.
Example 1-4a

12. Polygon Hierarchy Polygons Quadrilaterals Parallelograms Kites
Trapezoids
Isosceles Trapezoids
Rectangles
Rhombi
Squares

13. 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 bases Median = ½ (base + base)
Opposite sides parallel and congruent
Opposite angles congruent
Consecutive angles supplementary
Diagonals bisect each other
Rectangles
Rhombi
Isosceles Trapezoids
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

14. Homework
Homework:
pg ; (omit 17,18), 27-32, 35-38