Objectives Classify polygons based on their sides and angles. Find and use the measures of interior and exterior angles of polygons.

_________________– Each segment that forms a polygon. _________________– The common endpoint of two sides. _________________ – A segment that connects any two nonconsecutive vertices.

You can name a polygon by the number of its sides You can name a polygon by the number of its sides. The table shows the names of some common polygons. A polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints. Remember!

Example 1A: Identifying Polygons Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides.

Example 1B: Identifying Polygons Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides.

Example 1C: Identifying Polygons Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides.

Check It Out! Example 1D Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides.

Check It Out! Example 1E Tell whether the figure is a polygon. If it is a polygon, name it by the number of its sides.

Check It Out! Example 1F Tell whether the figure is a polygon. If it is a polygon, name it by the number of its sides.

All the sides are congruent in an equilateral polygon All the sides are congruent in an equilateral polygon. All the angles are congruent in an equiangular polygon. _______________– A polygon that is both equilateral and equiangular. _______________– A polygon that is not regular.

A polygon is: _______________- if any part of a diagonal contains points in the exterior of the polygon. _______________- if no diagonal contains points in the exterior. A regular polygon is always convex.

Example 2A: Classifying Polygons Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.

Check It Out! Example 2B Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.

Example 2C: Classifying Polygons Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.

Example 2D: Classifying Polygons Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.

Check It Out! Example 2E Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.

To find the sum of the interior angle measures of a convex polygon, draw all possible diagonals from one vertex of the polygon. This creates a set of triangles. The sum of the angle measures of all the triangles equals the sum of the angle measures of the polygon. By the Triangle Sum Theorem, the sum of the interior angle measures of a triangle is 180°. Remember!

In each convex polygon, the number of triangles formed is two less than the number of sides n. So the sum of the angle measures of all these triangles is (n — 2)180°.

Example 3A: Finding Interior Angle Measures and Sums in Polygons Find the sum of the interior angle measures of a convex heptagon.

Example 3B: Finding Interior Angle Measures and Sums in Polygons Find the measure of each interior angle of a regular 16-gon. Step 1 Find the sum of the interior angle measures. Step 2 Find the measure of one interior angle.

Example 3C: Finding Interior Angle Measures and Sums in Polygons Find the measure of each interior angle of pentagon ABCDE.

Check It Out! Example 3D Find the sum of the interior angle measures of a convex 15-gon.

Check It Out! Example 3E Find the measure of each interior angle of a regular decagon.

In the polygons below, an exterior angle has been measured at each vertex. Notice that in each case, the sum of the exterior angle measures is 360°.

An exterior angle is formed by one side of a polygon and the extension of a consecutive side. Remember!

Example 4A: Finding Interior Angle Measures and Sums in Polygons Find the measure of each exterior angle of a regular 20-gon.

Example 4B: Finding Interior Angle Measures and Sums in Polygons Find the value of b in polygon FGHJKL.

Check It Out! Example 4C Find the measure of each exterior angle of a regular dodecagon.

Check It Out! Example 4D Find the value of r in polygon JKLM.