Objectives Classify polygons based on their sides and

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 is a side of the polygon. The common endpoint of two sides is a vertex of the polygon. A segment that connects any two nonconsecutive vertices is a diagonal.

Remember! A polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints.

Number of sides • • • 3 4 5 6 7 8 9 10 11 12 Name of polygon Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon Undecagon Dodecagon

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

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

Example 1 C: Identifying Polygons Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides. not a polygon

Check It Out! Example 1 a Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides. not a polygon

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

Check It Out! Example 1 c Tell whether the figure is a polygon. If it is a polygon, name it by the number of its sides. not a polygon

equilateral polygon All the sides are congruent equiangular polygon All the angles are congruent regular polygon Both equilateral and equiangular *Note: If a polygon is not regular, it is called irregular.

concave – any part of a diagonal contains points in the exterior of the polygon convex - no diagonal contains points in the exterior Note: A regular polygon is always convex. Does that mean a irregular polygon is always concave?

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

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

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

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

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

Interior Angle Measures of CONVEX POLYGONS 1) Draw all possible diagonals from one vertex of the polygon. This creates a set of triangles.

Interior Angle Measures of CONVEX POLYGONS 1) Draw all possible diagonals from one vertex of the polygon. This creates a set of triangles. 2) The sum of the angle measures of all the triangles equals the sum of the angle measures of the polygon.

Remember! By the Triangle Sum Theorem, the sum of the interior angle measures of a triangle is 180°.

Example Find the sum of the interior angle measures of a convex heptagon. (n – 2)180° (7 – 2)180° 900°

Example Find the measure of each interior angle of a regular 16 -gon. Step 1 Find the sum of the interior angle measures. (n – 2)180° (16 – 2)180° = 2520° Step 2 Find the measure of one interior angle.

Your Turn Find the measure of each interior angle of a regular decagon. Step 1 Find the sum of the interior angle measures. (n – 2)180° (10 – 2)180° = 1440° Step 2 Find the measure of one interior angle.

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°.

Example Find the measure of each exterior angle of a regular 20 -gon. A 20 -gon has 20 sides and 20 vertices. sum of ext. s = 360°. measure of one ext. = The measure of each exterior angle of a regular 20 -gon is 18°.

Your Turn Find the measure of each exterior angle of a regular dodecagon. A dodecagon has 12 sides and 12 vertices. sum of ext. s = 360°. measure of one ext. The measure of each exterior angle of a regular dodecagon is 30°.

Lesson Quiz 1. Name the polygon by the number of its sides. Then tell whether the polygon is regular or irregular, concave or convex. nonagon; irregular; concave 2. Find the sum of the interior angle measures of a convex 11 -gon. 1620° 3. Find the measure of each interior angle of a regular 18 -gon. 160° 4. Find the measure of each exterior angle of a regular 15 -gon. 24°