Day 1 Properties of polygons Have paper, a pencil and calculator on your desk

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

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.

Example 1: Identifying Polygons Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides. Triangle Hexagon Octagon Pentagon N-gon Heptagon Nonagon Not a polygon polygon, hexagon

Example 2: Identifying Polygons Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides. polygon, heptagon

Example 3: 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 4 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 5 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 6 Tell whether the figure is a polygon. If it is a polygon, name it by the number of its sides. not a polygon

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 regular polygon is one that is both equilateral and equiangular. If a polygon is not regular, it is called irregular.

A polygon is concave if any part of a diagonal contains points in the exterior of the polygon. If no diagonal contains points in the exterior, then the polygon is convex. A regular polygon is always convex.

Example 7: Classifying Polygons Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. Regular, Concave Regular, Convex Irregular, Concave Irregular, Convex irregular, convex

Example 8: Classifying Polygons Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. irregular, concave

Example 9: Classifying Polygons Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. regular, 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.

The measure of each interior angle of a regular n-gon is :

Example 10: Finding Interior Angle Measures and Sums in Polygons Find the sum of the interior angle measures of a convex heptagon. (n – 2)180° Polygon  Sum Thm. (7 – 2)180° A heptagon has 7 sides, so substitute 7 for n. 900° Simplify.

Example 11: 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. (n – 2)180° Polygon  Sum Thm. Substitute 16 for n and simplify. (16 – 2)180° = 2520° Step 2 Find the measure of one interior angle. The int. s are , so divide by 16.

Example 12: Finding Interior Angle Measures and Sums in Polygons Find the Value of C Polygon  Sum Thm. (5 – 2)180° = 540° Polygon  Sum Thm. mA + mB + mC + mD + mE = 540° 35c + 18c + 32c + 32c + 18c = 540 Substitute. 135c = 540 Combine like terms. c = 4 Divide both sides by 135.

Example 12 Continued mA = 35(4°) = 140° mB = mE = 18(4°) = 72° mC = mD = 32(4°) = 128°

In the polygon below, an exterior angle has been measured at each vertex. Notice the sum of the exterior angle measures is 360°. For any polygon, the exterior angle and the interior angle at the same vertex must be a linear pair. So the interior angles measure to be: 180-41= 139° 180-55= 125° 180-110=70° 180-43=137° 180-111=69°

The measure of each exterior angle of a regular n-gon is: 360 n

Example 13: Finding Interior Angle Measures and Sums in Polygons 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°. Polygon  Sum Thm. A regular 20-gon has 20  ext. s, so divide the sum by 20. measure of one ext.  = The measure of each exterior angle of a regular 20-gon is 18°.

Example 14: Finding Interior Angle Measures and Sums in Polygons Find the value of b in polygon FGHJKL. Polygon Ext.  Sum Thm. 15b° + 18b° + 33b° + 16b° + 10b° + 28b° = 360° 120b = 360 Combine like terms. b = 3 Divide both sides by 120.

Lesson Quiz 1. Find the sum of the interior angle measures of a convex 11-gon. 2. Find the measure of each interior angle of a regular 18-gon. 3. Find the measure of each exterior angle of a regular 15-gon.

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