Explore 2 Exploring Interior Angles in Polygons To determine the sum of the interior angles for any polygon, you can use what you know about the Triangle Sum Theorem by considering how many triangles there are in other polygons. For example, by drawing the diagonal from a vertex of a quadrilateral, you can form two triangles. Since each triangle has an angle sum of 180, the quadrilateral must have an angle sum of 180 + 180 = 360. A Draw the diagonals from any one vertex for each polygon. Then state the number of triangles that are formed. The first two have already been completed. triangle quadrilateral quadrilateral 2 triangles 1 triangle 2 triangles 3 triangles B 4 triangles For each polygon, identify the number of sides and triangles, and determine the angle sums. Then complete the chart. The first two have already been done for you. Polygon Number of Sides 5 triangles 6 triangles Number of Triangles Sum of Interior Angle Measures Triangle 3 1 (1)180 = 180 Quadrilateral 4 2 (2)180 = 360 Pentagon 5 3 ( 3 ) 180 = 540 Houghton Mifflin Harcourt Publishing Company Hexagon 6 4 ( 4 ) 180 = 720 Decagon 10 8 ( 8 ) 180 = 1440 Module 7 314 Lesson 1
Do you notice a pattern between the number of sides and the number of triangles? If n represents the number of sides for any polygon, how can you represent the number of triangles? n - 2 Make a conjecture for a rule that would give the sum of the interior angles for any n-gon. Sum of interior angle measures = (n - 2) 180 Reflect 3. In a regular hexagon, how could you use the sum of the interior angles to determine the measure of each interior angle? Since the polygon is regular, you can divide the sum by 6 to determine each interior angle measure. 4. How might you determine the number of sides for a polygon whose interior angle sum is 3240? Write and solve an equation for n, where (n - 2) 180 = 3240. [# Explain 1 Using Interior Angles You can use the angle sum to determine the unknown measure of an angle of a polygon when you know the measures of the other angles. Polygon Angle Sum Theorem The sum of the measures of the interior angles of a convex polygon with n sides is (n 2)180. Example 1 Determine the unknown angle measures. For the nonagon shown, find the unknown angle measure x. First, use the Polygon Angle Sum Theorem to find the sum of the interior angles: n = 9 Houghton Mifflin Harcourt Publishing Company (n - 2)180 = (9-2)180 = (7)180 = 1260 Then solve for the unknown angle measure, x : 125 + 130 + 172 + 98 + 200 + 102 + 140 + 135 + x = 1260 x = 158 The unknown angle measure is 158. 98 102 135 140 200 x 125 130 172 Module 7 315 Lesson 1
B Determine the unknown interior angle measure of a convex octagon in which the measures of the seven other angles have a sum of 940. n = 8 8 Sum = ( - 2 ) 180 = ( ) 180 = 6 1080 940 + x = 1080 x = 140 The unknown angle measure is 140. Reflect 5. How might you use the Polygon Angle Sum Theorem to write a rule for determining the measure of each interior angle of any regular convex polygon with n sides? (n - 2) 180 You can divide the angle sum by n. n gives the measure of an interior angle for any regular polygon. Your Turn 6. Determine the unknown angle measures in this pentagon. x x n = 5 Sum = (5-2) 180 = (3) 180 = 540 270 + 2x = 540 2x = 270 x = 135 Each unknown angle measure is 135. 7. Determine the measure of the fourth interior angle of a quadrilateral if you know the other three measures are 89, 80, and 104. n = 4 Sum = (4-2) 180 = 2 (180 ) = 360 89 + 80 + 104 + x = 360 x = 87 The unknown angle measure is 87. 8. Determine the unknown angle measures in a hexagon whose six angles measure 69, 108, 135, 204, b, and 2b. n = 6 Sum = (6-2) 180 = (4) 180 = 720 b + 2b + 69 + 108 + 135 + 204 = 720 3b + 516 = 720 3b = 204 b = 68 2b = 136 The two unknown angle measures are 68 and 136. Houghton Mifflin Harcourt Publishing Company Module 7 316 Lesson 1
Explain 2 Proving the Exterior Angle Theorem An exterior angle is an angle formed by one side of a polygon and the extension of an adjacent side. Exterior angles form linear pairs with the interior angles. A remote interior angle is an interior angle that is not adjacent to the exterior angle. Remote interior angles Exterior angle Example 2 Follow the steps to investigate the relationship between each exterior angle of a triangle and its remote interior angles. Step 1 Use a straightedge to draw a triangle with angles 1, 2, and 3. Line up your straightedge along the side opposite angle 2. Extend the side from the vertex at angle 3. You have just constructed an exterior angle. The exterior angle is drawn supplementary to its adjacent interior angle. Step 2 You know the sum of the measures of the interior angles of a triangle. m 1 + m 2 + m 3 = 180 1 2 3 4 Since an exterior angle is supplementary to its adjacent interior angle, you also know: m 3 + m 4 = 180 Make a conjecture: What can you say about the measure of the exterior angle and the measures of its remote interior angles? Conjecture: The measure of the exterior angle is the same as the sum of the measures of its two remote interior angles. The conjecture you made in Step 2 can be formally stated as a theorem. Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. Houghton Mifflin Harcourt Publishing Company Step 3 Complete the proof of the Exterior Angle Theorem. 4 is an exterior angle. It forms a linear pair with interior angle 3. Its remote interior angles are 1 and 2. 1 3 4 By the Triangle Sum Theorem, m 1 + m 2 + m 3 = 180. Also, m 3 + m 4 = 180 because they are supplementary and make a straight angle. By the Substitution Property of Equality, then, m 1 + m 2 + m 3 = m 3 + m 4. Subtracting m 3 from each side of this equation leaves m 1 + m 2 = m 4. 2 This means that the measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. Module 7 317 Lesson 1
Explain 3 Using Exterior Angles You can apply the Exterior Angle Theorem to solve problems with unknown angle measures by writing and solving equations. Example 3 Determine the measure of the specified angle. Find m B. A Find m PRS. Q 2z D Write and solve an equation relating the exterior and remote interior angles. 145 = 2z + 5z - 2 145 = 7z - 2 z = 21 Now use this value for the unknown to evaluate the expression for the required angle. m B = (5z - 2) = (5(21) - 2) = (105-2) = 103 145 (5z - 2) C B R (3x - 8) S Write an equation relating the exterior and remote interior angles. 3x - 8 = (x + 2) + 90 Solve for the unknown. (x + 2) Use the value for the unknown to evaluate the expression for the required angle. P 3x - 8 = x + 92 2x = 100 x = 50 m PRS = (3x - 8) = (3 (50) - 8) = 142 Houghton Mifflin Harcourt Publishing Company Module 7 318 Lesson 1
Your Turn Determine the measure of the specified angle. 11. Determine m N in MNP. 12. If the exterior angle drawn measures 150, and the measure of D is twice that of E, find the N measure of the two remote interior angles. (3x + 7) D 150 E F G 63 (5x + 50) M P Q 5x + 50 = (3x + 7) + 63 5x + 50 = 3x + 70 2x = 20 x = 10 m N = (3x + 7) = (3(10) + 7) = 37 x + 2x = 150 3x = 150 x = 50 m E = x = 50 m D = 2x = 100
Name Class Date 7.2 Isosceles and Equilateral Triangles Essential Question: What are the special relationships among angles and sides in isosceles and equilateral triangles? Resource Locker Explore Investigating Isosceles Triangles An isosceles triangle is a triangle with at least two congruent sides. Vertex angle The congruent sides are called the legs of the triangle. Legs The angle formed by the legs is the vertex angle. The side opposite the vertex angle is the base. The angles that have the base as a side are the base angles. Base Base angles
Explain 1 Proving the Isosceles Triangle Theorem and Its Converse In the Explore, you made a conjecture that the base angles of an isosceles triangle are congruent. This conjecture can be proven so it can be stated as a theorem. Isosceles Triangle Theorem If two sides of a triangle are congruent, then the two angles opposite the sides are congruent. ifflin Harcourt Publishing Company This theorem is sometimes called the Base Angles Theorem and can also be stated as Base angles of an isosceles triangle are congruent. Module 7 328 Lesson 2
Explain 2 Proving the Equilateral Triangle Theorem and Its Converse An equilateral triangle is a triangle with three congruent sides. An equiangular triangle is a triangle with three congruent angles. Equilateral Triangle Theorem If a triangle is equilateral, then it is equiangular.
Explain 3 Using Properties of Isosceles and Equilateral Triangles You can use the properties of isosceles and equilateral triangles to solve problems involving these theorems. Example 3 Find the indicated measure. Katie is stitching the center inlay onto a banner that she created to represent her new tutorial service. It is an equilateral triangle with the following dimensions in centimeters. What is the length of each side of the triangle? A 6x - 5 B C 4x + 7 To find the length of each side of the triangle, first find the value of x. _ AC _ BC Converse of the Equilateral Triangle Theorem AC = BC 6x 5 = 4x + 7 Definition of congruence Substitution Property of Equality x = 6 Solve for x. Substitute 6 for x into either 6x 5 or 4x + 7. Houghton Mifflin Harcourt Publishing Company Image Credits: Nelvin C. Cepeda/ZUMA Press/Corbis 6 (6) 5 = 36 5 = 31 or 4 (6) + 7 = 24 + 7 = 31 So, the length of each side of the triangle is 31 cm. m T T 3x x R S To find the measure of the vertex angle of the triangle, first find the value of x. m R = m S = x Isosceles Triangle Theorem m R + m S + m T = 180 Triangle Sum Theorem x + x + 3x = 180 Substitution Property of Equality 5x = 180 Addition Property of Equality x = 36 Division Property of Equality 36 So, m T = 3x = 3( ) = 108. Module 7 331 Lesson 2
Your Turn 5. Find m P. P (3x + 3) Q (5x - 2) R m P = m Q = (3x + 3) 2 (3x + 3) + (5x - 2) = 180 x = 16 m P = (3x + 3) = (3(16) + 3) = 51 6. Katie s tutorial service is going so well that she is having shirts made with the equilateral triangle emblem. She has given the t-shirt company these dimensions. What is the length of each side of the triangle in centimeters? AB AC AB = AC 3_ 10 y + 9 = 4_ 5 y - 1 20 = y Therefore, _ 3 10 y + 9 = _ 3 (20) + 9 = 6 + 9 = 15 10 The length of each side is 15 cm. 3 y + 9 10 B A 4 y - 1 5 C