1 Tambourines The frame of the tambourine shown is a regular heptagon. What is the measure of each angle of the heptagon? Angles and Polygons 13.3 LESSON

2 Angles and Polygons 13.3 LESSON An interior angle of a polygon is an angle inside the polygon. You can find the measure of an interior angle of a regular polygon by dividing the sum of the measures of the interior angles by the number of sides.

3 Measures of Interior Angles of a Convex Polygon The sum of the measures of the interior angles of a convex n -gon is given by the formula (n – 2) 180˚. The measure of an interior angle of a regular n -gon is given by the formula. (n – 2) 180˚ n Angles and Polygons 13.3 LESSON

4 EXAMPLE 1 Finding the Sum of a Polygon’s Interior Angles Find the sum of the measures of the interior angles of the polygon. SOLUTION For a convex pentagon, n = 5. (n – 2) 180˚ = (5 – 2) 180˚ = 3 180˚ = 540˚ Angles and Polygons 13.3 LESSON

5 EXAMPLE 1 Finding the Sum of a Polygon’s Interior Angles Find the sum of the measures of the interior angles of the polygon. SOLUTION For a convex octagon, n = 8. (n – 2) 180˚ = (8 – 2) 180˚ = 6 180˚ = 1080˚ Angles and Polygons 13.3 LESSON

6 Find the measure of an interior angle of the frame of the heptagonal tambourine. EXAMPLE 2 Finding the Measure of an Interior Angle SOLUTION Because the tambourine is a regular heptagon, n = 7. Measure of an interior angle = (n – 2) 180˚ n = (7 – 2) 180˚ 7 ≈ 128.6˚ Write formula. Substitute 7 for n. Evaluate. Use a calculator. ANSWER The measure of an interior angle of the frame of the tambourine is about 128.6˚. Angles and Polygons 13.3 LESSON

7 Exterior Angles When you extend a side of a polygon, the angle that is adjacent to the interior angle is an exterior angle. In the diagram,  1 and  2 are exterior angles. An interior angle and an exterior angle at the same vertex form a straight angle. Angles and Polygons 13.3 LESSON

8 Find m  1 in the diagram. EXAMPLE 3 Finding the Measure of an Exterior Angle SOLUTION The angle that measures 87˚ forms a straight angle with  1, which is the exterior angle at the same vertex. m  1 = 93˚ Angles are supplementary. Subtract 87˚ from each side. m  ˚ = 180˚ Angles and Polygons 13.3 LESSON

9 Teapots The diagram shows a teapot in the shape of a regular hexagon. Find m  2. EXAMPLE 4 Finding an Angle Measure of a Regular Polygon SOLUTION Angles are supplementary. Substitute formula for m  1. m  1 + m  2 = 180˚ The measure of an interior angle of a regular hexagon is. (6 – 2) 180˚ 6 + m  2 = 180˚ (6 – 2) 180˚ 6 120˚ + m  2 = 180˚ m  2 = 60˚ Subtract 120˚ from each side. Simplify. Angles and Polygons 13.3 LESSON

10 Sum of Exterior Angle Measures Each vertex of a convex polygon has two exterior angles. If you draw one exterior angle at each vertex, then the sum of the measures of these angles is 360˚. The calculations below show that this is true for a triangle. m  4 + m  5 + m  6 = (180˚ – m  1) + (180˚ – m  2) + (180˚ – m  3) = (180˚ + 180˚ + 180˚) – (m  1 + m  2 + m  3) = 360˚= 540˚ – 180˚ Angles and Polygons 13.3 LESSON

11 Find the unknown angle measure in the diagram. EXAMPLE 5 Using the Sum of Measures of Exterior Angles SOLUTION Sum of measures of exterior angles of convex polygon is 360 ˚. x˚ + 81˚ + 100˚ + 106˚ = 360˚ x = 360 x = 73 Subtract 287 from each side. Add. ANSWER The angle measure is 73˚. Angles and Polygons 13.3 LESSON