1. Objectives
Define polygon, concave / convex polygon, and regular polygon
Find the sum of the measures of interior angles of a polygon
Find the sum of the measures of exterior angles of a polygon

2. Definition of polygon
A polygon is a closed plane figure formed by 3 or more sides that are line segments;
the segments only intersect at endpoints
no adjacent sides are collinear
Polygons are named using letters of consecutive vertices

3. Concave and Convex Polygons
A convex polygon has no diagonal with points outside the polygon
A concave polygon has at least one diagonal with points outside the polygon

4. Regular Polygon Definition
An equilateral polygon has all sides congruent
An equiangular polygon has all angles congruent
A regular polygon is both equilateral and equiangular
Note: A regular polygon is always convex

5. Sum of Interior Angles in Polygons
Convex Polygon
# of Sides
# of Triangles from 1 Vertex
Sum of Interior Angle Measures
Triangle 3 1
1* 180 = 180
Quadrilateral 4 2
2* 180 = 360
Pentagon 5
3* 180 = 540
Hexagon 6
4* 180 = 720
Heptagon 7
5* 180 = 900
Octagon 8
6* 180 = 1080
n-gon n
n – 2
(n – 2) * 180

6. Example: Sum of Interior Angles
Find m∠ X
Solution: The sum of the measures of the interior angles for a quadrilateral is (4 – 2) * 180 = 360
The marks in the illustration indicate that
m∠X = m∠Y. So the sum of all four interior angles is
m∠X + m∠X = 360
2 m∠X = 360
2 m∠X = 170
m∠X = 85

7. Polygon Exterior Angle Sum Theorem
The sum of the measures of the exterior angles of a polygon, one at each vertex is 360.
m∠1 + m∠2 + m∠3 + m∠4 + m∠5 = 360

8. Example: Exterior Angle Sum
What is the measure of an interior angle of a regular octagon?
Solution:
8 * exterior angle = 360 (Ext. Angle Sum)
exterior angle = 45
interior angle = 180 – exterior angle
interior angle = 180 – 45 = 135