Vocabulary side of a polygon vertex of a polygon diagonal regular polygon concave/convex interior/exterior angles (i , e)

Each segment that forms a polygon is a side of the polygon 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. Where are interior/exterior angles? (i , e)

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 1: Identifying Polygons Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides. polygon, hexagon polygon, heptagon 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 2: Classifying Polygons irregular, convex regular, convex irregular, concave regular, convex

To find the sum of the interior angle measures (Si) 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. Notice, there are (n – 2) triangles.

Convex Polygons Si = (n-2) 180

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°. Se = 360

REGULAR POLYGONS REGULAR: me = 360/n OR n = 360/me Also, you can use Si = n(i) for Regular For REGULAR polygons always work with n, i , e Remember, for any convex polygon: mi+me = 180, Si = (n-2) 180, Se = 360

Example 3: Finding Interior Angle Measures and Sums in Polygons Find the measure of each interior angle of pentagon ABCDE. Si = (n-2) 180 n = 5 Polygon  Sum Thm. (5 – 2)180° = 540° 35c + 18c + 32c + 32c + 18c = 540 Substitute. 135c = 540 Combine like terms. c = 4 Divide both sides by 135.

Example 3 Continued mA = 35(4°) = 140° mB = mE = 18(4°) = 72° mC = mD = 32(4°) = 128°

Example 3B: Finding Interior Angle Measures and Sums in Polygons Find the measure of each interior angle of a regular 18-gon. Then, find the sum of the angles. Regular: 360/n = me → me= 20 mi + me = 180 → mi = 160 Si = n(i) → Si = 2880

Check It Out! Find the measure of each interior angle of a regular decagon. me = 36, mi = 144 If an interior angle of a regular polygon is 135, how many sides does it have? me = 45, n = 8 If the sum of the interior angles is 2880, how many sides does it have? 2880 = (n-2)180, n = 18 4. Can a polygon have Si = 3080? 3080 = (n-2)180, n = 19.111, NO!

READ CAREFULLY: An interior or each interior versus sum of interior…. An exterior or each exterior versus sum of the exterior… Regular or irregular…

Lesson Quiz 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. 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.