Section 6.1 Angles of Polygons

A diagonal of a polygon is a segment that connects any two nonconsecutive vertices. The vertices of polygon PQRST that are not consecutive with vertex P are vertices R and S. Therefore, polygon PQRST has two diagonals from vertex P, PR and PS. Notice that the diagonals from vertex P separate the polygon into three triangles. The sum of the angle measures of a polygon is the sum of the angle measures of the triangles formed by drawing all the possible diagonals from one vertex.

Since the sum of the angle measures of a triangle is 180, we can make a table and look for a pattern to find the sum of the angle measures for any convex polygon. This leads to the following theorem: You can use the Polygon Interior Angles Sum Theorem to find the sum of the interior angles of a polygon and to find missing measures in polygons.

Example 1: a) Find the sum of the measures of the interior angles of a convex nonagon. A nonagon has nine sides. Use the Polygon Interior Angles Sum Theorem to find the sum of its interior angle measures. (n – 2) ● 180 = (9 – 2) ● 180 n = 9 = 7 ● 180 or 1260 Simplify. Answer: The sum of the measures is 1260°.

Example 1: b) Find the measure of each interior angle of parallelogram RSTU. Step 1 Find the value of x. Since n = 4 the sum of the measures of the interior angles is 180(4 – 2) or 360°. Write an equation to express the sum of the measures of the interior angles of the polygon. 360 = mÐR + mÐS + mÐT + mÐU Sum of measures of interior angles 360 = 5x + (11x + 4) + 5x + (11x + 4) Substitution 360 = 32x + 8 Combine like terms 352 = 32x Subtract 8 from each side 11 = x Divide each side by 32

Step 2 Use the value of x to find the measure of each angle. mR = 5x = 5(11) or 55° mS = 11x + 4 = 11(11) + 4 or 125° mT = 5x = 5(11) or 55° mU = 11x + 4 = 11(11) + 4 or 125° Answer: mR = 55°, mS = 125°, mT = 55°, mU = 125°

Example 3: 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 measure of one of the interior angles of the pentagon. Find the sum of the interior angle measures. (n – 2) ● 180 = (5 – 2) ● 180 n = 5 = 3 ● 180 or 540 Simplify. Find the measure of one interior angle. Substitution = 108° Divide.

Given the interior angle measure of a regular polygon, you can also use the Polygon Interior Angles Sum Theorem to find a polygon’s number of sides. Example 3: a) The measure of an interior angle of a regular polygon is 150. 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. S = 180(n – 2) Interior Angle Sum Theorem (150)n = 180(n – 2) S = 150n 150n = 180n – 360 Distributive Property 0 = 30n – 360 Subtract 150n from each side. 360 = 30n Add 360 to each side. 12 = n Divide each side by 30.

Example 3: b) The measure of an interior angle of a regular polygon is 144. 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. S = 180(n – 2) Interior Angle Sum Theorem (144)n = 180(n – 2) S = 144n 144n = 180n – 360 Distributive Property 0 = 36n – 360 Subtract 144n from each side. 360 = 36n Add 360 to each side. 10 = n Divide each side by 36.

Does a relationship exist between the number of sides a convex polygon and the sum of its exterior angle measures? Examine the polygons below in which an exterior angle has been measured at each vertex Did you notice that the sum of the exterior angle measures in each case is 360? This suggests the following theorem:

a) Find the value of x in the diagram. Example 4: a) Find the value of x in the diagram. 5x + (4x – 6) + (5x – 5) + (4x + 3) + (6x – 12) + (2x + 3) + (5x + 5) = 360 (5x + 4x + 5x + 4x + 6x + 2x + 5x) + [(–6) + (–5) + 3 + (–12) + 3 + 5] = 360 31x – 12 = 360 31x = 372 x = 12

Example 4: b) Find the measure of each exterior angle of a regular decagon. A regular decagon has 10 congruent sides and 10 congruent angles. The exterior angles are also congruent, since angles supplementary to congruent angles are congruent. Let n = the measure of each exterior angle and write and solve an equation. 10n = 360 Polygon Exterior Angle Sum Theorem n = 36 Divide each side by 10.