Section 3-5: The Polygon Angle-Sum Theorem

Objectives To classify polygons. To find the sums of the measures of the interior and exterior angles of a polygon.

Vocabulary Polygon Convex Polygon Concave Polygon Equilateral Polygon Equiangular Polygon Regular Polygon

Polygon A polygon is a closed plane figure with at least three sides that are segments. The sides intersect only at their endpoints and no adjacent sides are collinear.

Are the following figures polygons?

Naming a Polygon

Classification of Polygons by Sides Number of SidesName of Figure 3Triangle 4Quadrilateral 5Pentagon 6Hexagon 8Octagon 9Nonagon 10Decagon 12Dodecagon nn-gon

Convex Polygon A convex polygon has no diagonal with points outside the polygon.

Concave Polygon A concave polygon has at least one diagonal with points outside the polygon.

Classify each Polygon Classify each polygon by its sides. Identify the polygon as convex or concave.

Theorem 3-14: “Polygon Angle Sum Theorem” The measures of the angles of an n-gon is:

Finding a Polygon Angle Sum Find the sum of the measures of the angles of a 15-gon.

Using the Polygon Angle-Sum Theorem Solve for x.

Theorem 3-15: “Polygon Exterior Angle-Sum Theorem” The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360º.

Equilateral Polygon An equilateral polygon has all sides congruent.

Equiangular Polygon An equiangular polygon has all angles congruent.

Regular Polygon A regular polygon is both equilateral and equiangular.

Example Find the measure of an interior angle and an exterior angle of a regular octagon.