Classify polygons based on their sides and angles.

Find and use the measures of interior and exterior angles of polygons.

Objectives

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.

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

Remember!

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 1A: Identifying Polygons

Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides.

polygon, hexagon

Example 1B: Identifying Polygons

Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides.

polygon, heptagon

Example 1C: 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 1a

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 1b

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 1c

Tell whether the figure is a polygon. If it is a polygon, name it by the number of its sides.

not a polygon

equilateral polygonAll the sides are congruent

equiangular polygonAll 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 2A: Classifying Polygons

Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.

irregular, convex

Example 2B: Classifying Polygons

Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.

irregular, concave

Example 2C: Classifying Polygons

Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.

regular, convex

Check It Out! Example 2a

Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.

regular, convex

Check It Out! Example 2b

Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.

irregular, concave

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.

Interior Angle Measures of CONVEX POLYGONS

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

Remember!

Find the sum of the interior angle measures of a convex heptagon.

(n – 2)180°

(7 – 2)180°

900°

Example

Example

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.

(n – 2)180°

(16 – 2)180° = 2520°

Find the measure of each interior angle of a regular decagon.

Step 1 Find the sum of the interior angle measures.

Step 2 Find the measure of one interior angle.

Your Turn

(n – 2)180°

(10 – 2)180° = 1440°

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°.

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°.

The measure of each exterior angle of a regular 20-gon is 18°.

measure of one ext. =

Example

Find the measure of each exterior angle of a regular dodecagon.

A dodecagon has 12 sides and 12 vertices.

sum of ext. s = 360°.

The measure of each exterior angle of a regular dodecagon is 30°.

measure of one ext.

Your Turn

1. Name the polygon by the number of its sides. Then tell whether the polygon is regular or irregular, concave or convex.

2. Find the sum of the interior angle measures of a convex 11-gon.

Lesson Quiz

nonagon; irregular; concave

1620°

3. Find the measure of each interior angle of a regular 18-gon.

4. Find the measure of each exterior angle of a regular 15-gon.

160°

24°