1. 3.5 The Polygon Angle-Sum Theorems
Geometry
Mr. Barnes

2. Objectives: To Classify Polygons
To find the sums of the measures of the interior and exterior angles of polygons.

3. Definitions:
SIDE
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.

4. Example 1: Identifying Polygons
State whether the figure is a polygon.
If it is not, explain why.
Not D- because 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.
Figures A, B, and C are polygons.

5. Polygons are named by the number of sides they have – MEMORIZE
Type of Polygon 3
Triangle 4
Quadrilateral 5
Pentagon 6
Hexagon 7
Heptagon

6. Polygons are named by the number of sides they have – MEMORIZE
Type of Polygon 8
Octagon 9
Nonagon 10
Decagon 12
Dodecagon n
n-gon

7. Convex or concave?
Convex if no line that contains a side of the polygon contains a point in the interior of the polygon.
Concave or non-convex if a line does contain a side of the polygon containing a point on the interior of the polygon.
See how it doesn’t go on the
Inside-- convex
See how this crosses
a point on the inside?
Concave.

8. Convex or concave? CONCAVE CONVEX
Identify the polygon and state whether it is convex or concave.
CONCAVE
A polygon is EQUILATERAL
If all of its sides are congruent.
A polygon is EQUIANGULAR
if all of its interior angles are congruent.
A polygon is REGULAR if it is
equilateral and equiangular.
CONVEX

9. Ex. : Interior Angles of a Quadrilateral
80°
Ex. : Interior Angles of a Quadrilateral
70°
2x° x°
x°+ 2x° + 70° + 80° = 360°
3x = 360
3x = 210
x = 70
Sum of the measures of int. s of
A quadrilateral is 360°
Combine like terms
Subtract 150 from each side.
Divide each side by 3.
Find m Q and mR.
mQ = x° = 70°
mR = 2x°= 140°
►So, mQ = 70° and mR = 140°

10. Investigation Activity
Sketch polygons with 4, 5, 6, 7, and 8 sides
Divide Each Polygon into triangles by drawing all diagonals that are possible from one vertex
Multiply the number of triangles by 180 to find the sum of the measures of the angles of each polygon.
Look for a pattern. Describe any that you have found.
Write a rule for the sum of the measures of the angles of an n-gon

11. Polygon Angle-Sum Theorem
The sum of the measures of the angles of an n-gon is
(n-2)180
Ex: Find the sum of the measures of the angles of a 15-gon
Sum = (n-2)180
= (15-2)180
= (13)180
= 2340

12. Example
The sum of the interior angles of a polygon is How many sides does the polygon have?
Sum = (n-2)180
9180 = (n-2)180
51 = n-2
53 = n
The polygon has 53 sides.

13. Polygon Exterior Angle-Sum Theorem
The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360.
An equilateral polygon has all sides congruent
An equiangular polygon has all angles congruent
A regular polygon is both equilateral and equiangular.