Splash Screen. Five-Minute Check (over Chapter 5) NGSSS Then/Now New Vocabulary Theorem 6.1: Polygon Interior Angles Sum Example 1: Find the Interior Angles Sum of a Polygon Example 2: Real-World Example: Interior Angle Measure of Regular Polygon
Theorem 6.1: Polygon Interior Angles Sum
Example 1: Find the Interior Angles Sum of a Polygon
Example 2: Real-World Example: Interior Angle Measure of Regular Polygon
Example 3: Find Number of Sides Given Interior Angle Measure
Theorem 6.2: Polygon Exterior Angles Sum
Example 4: Find Exterior Angle Measures of a PolygonLesson Menu
MA.912.G.2.2Determine the measures of interior and exterior angles of polygons, justifying the method used.
MA.912.G.3.4 Prove theorems involving quadrilaterals.NGSSS
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 1A
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.
Find the Interior Angles Sum of a Polygon
B. Find the measure of each interior angle of parallelogram RSTU.
Step 1 Find x.Example 1B
Understand Look at the diagram of the situation. The measure of the angle of a corner in between two walkways is the interior angle of a regular pentagon.
Plan Use the Polygon Interior Angles Sum Theorem to find the sum of the measures of the angles. Since the angles of a regular polygon are congruent, divide this sum by the number of angles to find the measure of each interior angle.Example 2
Answer: The measure of one of the interior angles of the food court is 108.
Check To verify that this measure is correct, use a ruler and a protractor to draw a regular pentagon using 108 as the measure of each interior angle. The last side drawn should connect with the beginning point of the first segment drawn.Example 2
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.Example 3
Use the Polygon Exterior Angles Sum Theorem to write an equation. Then solve for x.
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
Answer: x = 12Example 4A
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.
Answer: The measure of each exterior angle of a regular decagon is 36.Example 4B