8.1 Angles of Polygons

Objectives Find the sum of the measures of the interior angles of a polygon Find the sum of the measures of the exterior angles of a polygon

Sum of the Measures of the Interior Angles of a Polygon We have already learned the name of a polygon depends on the number of sides in the polygon: triangle, quadrilateral, pentagon, hexagon, and so forth. The sum of the measures of the interior angles of a polygon also depends on the number of sides.

Sum of the Measures of the Interior Angles of a Polygon From a previous lesson we learned the sum of the measures of the interior angles of a quadrilateral are known by dividing the quadrilateral into two triangles. You can use this triangle method to find the sum of the measures of the interior angles of any convex polygon with n sides, called an n - gon.

Sum of the Measures of the Interior Angles of a Polygon # of sides # of triangles Sum of measures of interior ’s Triangle 3 1 1 ● 180 = 180 Quadrilateral 2 ● 180 = 360 Pentagon Hexagon Nonagon (9) n - gon n

Sum of the Measures of the Interior Angles of a Polygon From the previous slide, we have discovered that the sum of the measures of the interior angles of a convex n - gon is (n – 2) ● 180 This relationship can be used to find the measure of each interior angle in a regular n - gon because the angles are all congruent.

Interior Angle Sum Theorem Interior Angle Sum Theorem If a convex polygon has n sides and S is the sum of its interior angles, then S = 180(n – 2).

Example 1: ARCHITECTURE A mall is designed so that five walkways meet at a food court that is in the shape of a regular pentagon. Find the sum of measures of the interior angles of the pentagon. Since a pentagon is a convex polygon, we can use the Angle Sum Theorem.

Example 1: Interior Angle Sum Theorem Simplify. Answer: The sum of the measures of the angles is 540.

Your Turn: A decorative window is designed to have the shape of a regular octagon. Find the sum of the measures of the interior angles of the octagon. Answer: 1080

Example 2: The measure of an interior angle of a regular polygon is 135. Find the number of sides in the polygon. Use the Interior Angle Sum Theorem to write an equation to solve for n, the number of sides. Interior Angle Sum Theorem Distributive Property Subtract 135n from each side. Add 360 to each side. Divide each side by 45. Answer: The polygon has 8 sides.

Your Turn: The measure of an interior angle of a regular polygon is 144. Find the number of sides in the polygon. Answer: The polygon has 10 sides.

Example 3: Find the measure of each interior angle. Since the sum of the measures of the interior angles is Write an equation to express the sum of the measures of the interior angles of the polygon.

Example 3: Sum of measures of angles Substitution Combine like terms. Subtract 8 from each side. Divide each side by 32.

Example 3: Use the value of x to find the measure of each angle. Answer:

Your Turn: Find the measure of each interior angle. Answer:

Sum of the Measures of the Exterior Angles of a Polygon Interestingly, the measures of the exterior angles of a polygon is an even easier formula. Let’s look at the following example to understand it.

Exterior Angle Sum Theorem Exterior Angle Sum Theorem If a polygon is convex, then the sum of the measures of the exterior angles, one at each vertex, is 360°.

Example 4: Find the measures of an exterior angle and an interior angle of convex regular nonagon ABCDEFGHJ. At each vertex, extend a side to form one exterior angle.

Example 4: The sum of the measures of the exterior angles is 360. A convex regular nonagon has 9 congruent exterior angles. Divide each side by 9. Answer: The measure of each exterior angle is 40. Since each exterior angle and its corresponding interior angle form a linear pair, the measure of the interior angle is 180 – 40 or 140.

Your Turn: Find the measures of an exterior angle and an interior angle of convex regular hexagon ABCDEF. Answer: 60; 120

Assignment Pre-AP Geometry: Pg. 407 #13 - 41