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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

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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

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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

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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

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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 3* 180 = 540

Hexagon 6 4 4* 180 = 720

Heptagon 7 5 5* 180 = 900

Octagon 8 6 6* 180 = 1080

n-gon n n – 2 (n – 2) * 180

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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 + 100 + 90 = 3602 m∠X + 190 = 3602 m∠X = 170 m∠X = 85

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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

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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