LESSON 6–1 Angles of Polygons

Five-Minute Check (over Chapter 5) TEKS 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 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 Polygon Lesson Menu

State whether this sentence is always, sometimes, or never true State whether this sentence is always, sometimes, or never true. The three altitudes of a triangle intersect at a point inside the triangle. A. always B. sometimes C. never 5-Minute Check 1

Find n and list the sides of ΔPQR in order from shortest to longest if mP = 12n – 15, mQ = 7n + 26, and mR = 8n – 47. A. n = 8; B. n = 8; C. n = 6; D. n = 6; 5-Minute Check 2

State the assumption you would make to start an indirect proof of the statement. If –2x ≥ 18, then x ≤ –9. A. x is positive. B. –2x < 18 C. x > –9 D. x < 9 5-Minute Check 3

Find the range for the measure of the third side of a triangle given that the measures of two sides are 43 and 29. A. n > 14 B. 14 < n < 72 C. 29 < n < 43 D. n > 0 5-Minute Check 4

Write an inequality relating mABD and mCBD. A. mABD < mCBD B. mABD ≤ mCBD C. mABD > mCBD D. mABD = mCBD 5-Minute Check 5

Write an equation that you can use to find the measures of the angles of the triangle. A. x – 5 + 3x = 180 B. x – 5 + 3x + 111 = 180 C. x – 5 + 3x = 69 D. x – 5 + 3x = 111 5-Minute Check 6

Mathematical Processes G.1(A), G.1(G) Targeted TEKS G.5(A) Investigate patterns to make conjectures about geometric relationships, including angles formed by parallel lines cut by a transversal, criteria required for triangle congruence, special segments of triangles, diagonals of quadrilaterals, interior and exterior angles of polygons, and special segments and angles of circles choosing from a variety of tools. Mathematical Processes G.1(A), G.1(G) TEKS

You named and classified polygons. Find and use the sum of the measures of the interior angles of a polygon. Find and use the sum of the measures of the exterior angles of a polygon. Then/Now

diagonal Vocabulary

Concept 1

Answer: The sum of the measures is 1260. Find the Interior Angles Sum of a Polygon 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

B. Find the measure of each interior angle of parallelogram RSTU. Find the Interior Angles Sum of a Polygon B. Find the measure of each interior angle of parallelogram RSTU. Step 1 Find x. 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 1B

Sum of measures of interior angles Find the Interior Angles Sum of a Polygon Sum of measures of interior angles Substitution Combine like terms. Subtract 8 from each side. Divide each side by 32. Example 1B

Step 2 Use the value of x to find the measure of each angle. Find the Interior Angles Sum of a Polygon 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 1B

A. Find the sum of the measures of the interior angles of a convex octagon. B. 1080 C. 1260 D. 1440 Example 1A

B. Find the value of x. A. x = 7.8 B. x = 22.2 C. x = 15 D. x = 10 Example 1B

Interior Angle Measure of Regular Polygon 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. Example 2

Interior Angle Measure of Regular Polygon Analyze 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. Formulate 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

Determine Find the sum of the interior angle measures. Interior Angle Measure of Regular Polygon Determine 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 Divide. Example 2

Interior Angle Measure of Regular Polygon Answer: The measure of one of the interior angles of the food court is 108. Justify 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. Evaluate The diagram of a regular pentagon models the food court in the problem. Diagrams can help you visualize three-dimensional problems to make them easier to solve. Example 2

A pottery mold makes bowls that are in the shape of a regular heptagon A pottery mold makes bowls that are in the shape of a regular heptagon. Find the measure of one of the interior angles of the bowl. A. 130° B. 128.57° C. 140° D. 125.5° Example 2

S = 180(n – 2) Interior Angle Sum Theorem (150)n = 180(n – 2) S = 150n Find Number of Sides Given Interior Angle Measure 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

Answer: The polygon has 12 sides. Find Number of Sides Given Interior Angle Measure 360 = 30n Add 360 to each side. 12 = n Divide each side by 30. Answer: The polygon has 12 sides. Example 3

The measure of an interior angle of a regular polygon is 144 The measure of an interior angle of a regular polygon is 144. Find the number of sides in the polygon. A. 12 B. 9 C. 11 D. 10 Example 3

Concept 2

A. Find the value of x in the diagram. Find Exterior Angle Measures of a Polygon A. Find the value of x in the diagram. Example 4A

Find Exterior Angle Measures of a Polygon 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 = 12 Example 4A

B. Find the measure of each exterior angle of a regular decagon. Find Exterior Angle Measures of a Polygon 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

A. Find the value of x in the diagram. B. 12 C. 14 D. 15 Example 4A

B. Find the measure of each exterior angle of a regular pentagon. Example 4B

LESSON 6–1 Angles of Polygons