# Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition.

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1 Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition. 1. measures less than By the Exterior Angle Inequality Theorem, the exterior angle ( ) is larger than either remote interior angle ( and ). Also,, and. By substitution,. Therefore, must be larger than each individual angle. By the Exterior Angle Inequality Theorem, 2. measures greater than m 7 By the Exterior Angle Inequality Theorem, the exterior angle ( 5) is larger than either remote interior angle ( 7 and 8). Similarly, the exterior angle ( 9) is larger than either remote interior angle ( 7 and 6). Therefore, 5 and 9 are both larger than. 3. measures greater than m 2 By the Exterior Angle Inequality Theorem, the exterior angle ( 4) is larger than either remote interior angle ( 1 and 2). Therefore, 4 is larger than. 4. measures less than m 9 By the Exterior Angle Inequality Theorem, the exterior angle ( ) is larger than either remote interior angle ( and ). Also,, and (. By substitution,. Therefore, must be larger than each individual angle so is larger than both and. esolutions Manual - Powered by Cognero Page 1

2 List the angles and sides of each triangle in order from smallest to largest. 5. Here it is given that,. Therefore, from angle-side relationships in a triangle, we know that Angle: Side: 6. By the Triangle Sum Theorem, So, From angle-side relationships in a triangle, we know Angle: J, K, L Side: 7. HANG GLIDING The supports on a hang glider form triangles like the one shown. Which is longer the support represented by or the support represented by? Explain your reasoning. Since the angle across from segment is larger than the angle across from, we can conclude that is longer than due to Theorem 5.9, which states that if one side of a triangle is longer than another, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side. esolutions Manual - Powered by Cognero Page 2

3 Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition. 8. measures greater than m 2 By the Exterior Angle Inequality Theorem, the exterior angle ( 4) is larger than either remote interior angle ( 1 and 2). Therefore, m 4 > m measures less than m 4 By the Exterior Angle Inequality Theorem, the exterior angle ( 4) is greater than either remote interior angle ( 1 and 2).Therefore, m 1< m 4 and m 2< m measures less than m 5 By the Exterior Angle Inequality Theorem, the exterior angle ( 5) is larger than either remote interior angle ( 7 and 8). Therefore, m 7 < m 5 and m 8 < m measures less than m 9 By the Exterior Angle Inequality Theorem, the exterior angle ( 9) is larger than either remote interior angle from two different triangles ( 6 and 7 from one triangle and and from another. Therefore,,,, and. 12. measures greater than m 8 By the Exterior Angle Inequality Theorem, the exterior angle ( 2) is larger than either remote interior angle ( 6 and 8). Similarly, the exterior angle ( 5) is larger than either remote interior angle ( 7 and 8). Therefore, m 2 > m 8 and m 5 > m measures greater than m 7 By the Exterior Angle Inequality Theorem, the exterior angle ( 9) is larger than either remote interior angle ( 6 and 7). Similarly, the exterior angle ( 5) is larger than either remote interior angle ( 7 and 8). Therefore, m 9 > m 7 and m 5 > m 7 esolutions Manual - Powered by Cognero Page 3

4 List the angles and sides of each triangle in order from smallest to largest. 14. The hypotenuse of the right triangle must be greater than the other two sides. Therefore, By Theorem 5.9, the measure of the angle opposite the longer side has a greater measure tahn the angle opposite the shorter side, therefore Angle: Side: 15. Based on the diagram, we see that By Theorem 5.9, the measure of the angle opposite the longer side has a greater measure than the angle opposite the shorter side, therefore Angle: Side: 16. Therefore, by Theorem 5.10 we know that the side opposite the greater angle is longer than the side opposite a lesser angle and Angle: Side: esolutions Manual - Powered by Cognero Page 4

5 17.. Therefore, by Theorem 5.10 we know that the side opposite the greater angle is longer than the side opposite a lesser angle and Angle: Side: 18. By the Triangle Sum Theorem, Therefore, by Theorem 5.10 we know that the side opposite the greater angle is longer than the side opposite a lesser angle and Angle: Side: 19. By the Triangle Sum Theorem, So, Therefore, by Theorem 5.10 we know that the side opposite the greater angle is longer than the side opposite a lesser angle and Angle: Side: esolutions Manual - Powered by Cognero Page 5

6 20. SPORTS Ben, Gilberto, and Hannah are playing Ultimate. Hannah is trying to decide if she should pass to Ben or Gilberto. Which player should she choose in order to have the shorter passing distance? Explain your reasoning. Based on the Triangle Sum Theorem, the measure of the missing angle next to Ben is degrees. Since 48 < 70, the side connecting Hannah to Ben is the shortest, based on Theorem Therefore, Hannah should pass to Ben. 21. RAMPS The wedge below represents a bike ramp. Which is longer, the length of the ramp or the length of the top surface of the ramp? Explain your reasoning using Theorem 5.9. If m X = 90, then, based on the Triangle Sum Theorem, m Y + m Z = 90, so m Y < 90 by the definition of inequality. So m X > m Y. According to Theorem 5.10, if m X > m Y, then the length of the side opposite X must be greater than the length of the side opposite Y. Since is opposite X, and is opposite Y, then YZ > XZ. Therefore YZ, the length of the top surface of the ramp, must be greater than the length of the ramp. esolutions Manual - Powered by Cognero Page 6

7 List the angles and sides of each triangle in order from smallest to largest. 22. Using the Triangle Sum Theorem, we can solve for x, as shown below. degrees and the degrees. Therefore,. By Theorem 5.10, we know that the lengths of sides across from larger angles are longer than those across from shorter angles so. Angle: Side: 23. Using the Triangle Sum Theorem, we can solve for x, as shown below. degrees, degrees and the degrees. Therefore,. By Theorem 5.10, we know that the lengths of sides across from larger angles are longer than those across from shorter angles so. Angle: P, Q, M Side: esolutions Manual - Powered by Cognero Page 7

8 Use the figure to determine which angle has the greatest measure , 5, 6 By the Exterior Angle Inequality, we know that the measure of the exterior angle of a triangle is greater than the measure of either of its corresponding remote interior angles. Therefore, since is the exterior angle and and are its remote interior angles, the m 1 is greater than m 5 and m , 4, 6 By the Exterior Angle Inequality, we know that the measure of the exterior angle of a triangle is greater than the measure of either of its corresponding remote interior angles. Therefore, since is the exterior angle and and are its remote interior angles, the m 2 is greater than m 4 and m , 4, 5 By the Exterior Angle Inequality, we know that the measure of the exterior angle of a triangle is greater than the measure of either of its corresponding remote interior angles. Therefore, since is the exterior angle and and are its remote interior angles, the m 7 is greater than m 4 and m , 11, 12 By the Exterior Angle Inequality, we know that the measure of the exterior angle of a triangle is greater than the measure of either of its corresponding remote interior angles. Therefore, since is the exterior angle and and are its remote interior angles, the m 3 is greater than m 11 and m , 9, 14 By the Exterior Angle Inequality, we know that the measure of the exterior angle of a triangle is greater than the measure of either of its corresponding remote interior angles. Therefore, since is the exterior angle and and the sum of and are its remote interior angles, the m 3 is greater than m 9 and the sum of the m 11 and m 14. If is greater than the sum of and, then it is greater than the measure of the individual angles. Therefore, the m 3 is greater than m 9 and m 14. esolutions Manual - Powered by Cognero Page 8

9 29. 8, 10, 11 By the Exterior Angle Inequality, we know that the measure of the exterior angle of a triangle is greater than the measure of either of its corresponding remote interior angles. Therefore, since is the exterior angle and and the sum of and are its remote interior angles, the is greater than m 10 and the sum of the m 11 and m 14. If is greater than the sum of and, then it is greater than the measure of the individual angles. Therefore, the m 8 is greater than m 10 and m 11. Use the figure to determine the relationship between the measures of the given angles. 30. ABD, BDA The side opposite is, which is of length 13. The side opposite is, which is of length 3. Since by Theorem BCF, CFB The side opposite is, which is of length 15. The side opposite is, which is of length 14. Since by Theorem BFD, BDF The side opposite is, which is of length 12. The side opposite is, which is of length 15. Since by Theorem DBF, BFD The side opposite is, which is of length 5. The side opposite is, which is of length 12. Since by Theorem 5.9. esolutions Manual - Powered by Cognero Page 9

10 Use the figure to determine the relationship between the lengths of the given sides. 34. SM, MR Since are a linear pair, The side opposite is. The side opposite is. In,, since 60 < 70. Therefore, by Theorem 5.10,. 35. RP, MP Since form a straight angle,. The side opposite is. The side opposite is. In,, since 70 < 35. Therefore, by Theorem 5.10,. 36. RQ, PQ The side opposite is. The side opposite is. In,, since 30 < 85. Therefore, by Theorem 5.10, 37. RM, RQ By the Triangle Sum Theorem,. The side opposite PQR is. The side opposite RPQ is. In, m PQR > m RPQ, since 65 > 30. Therefore, by Theorem 5.10, PR > RQ. Also, form a straight angle, and, by the Triangle Sum Theorem,. The side opposite MPR is. The side opposite PMR is. In, m MPR > m PMR, since 75 > 70. Therefore, by Theorem 5.10,. If and, then by the transitive property of inequality, Thus, esolutions Manual - Powered by Cognero Page 10

11 38. HIKING Justin and his family are hiking around a lake as shown in the diagram. Order the angles of the triangle formed by their path from largest to smallest. The side opposite is of length 0.5. The side opposite is of length 0.4. The side opposite is of length Since 0.5 > 0.45 > 0.4, by Theorem 5.9, m 3 > m 1 > m 2. COORDINATE GEOMETRY List the angles of each triangle with the given vertices in order from smallest to largest. Justify your answer. 39. A( 4, 6), B( 2, 1), C(5, 6) Use the Distance Formula to find the side lengths. So, Then, esolutions Manual - Powered by Cognero Page 11

12 40. X( 3, 2), Y(3, 2), Z( 3, 6) Use the Distance Formula to find the side lengths. So, Then, 41. List the side lengths of the triangles in the figure from shortest to longest. Explain your reasoning. AB, BC, AC, CD, BD; In, AB < BC < AC and in, BC < CD < BD. By the figure AC < CD, so BC < AC < CD. 42. MULTIPLE REPRESENTATIONS In this problem, you will explore the relationship between the sides of a triangle. a. GEOMETRIC Draw three triangles, including one acute, one obtuse, and one right angle. Label the vertices of each triangle A, B, and C. b. TABULAR Measure the length of each side of the three triangles. Then copy and complete the table. c. TABULAR Create two additional tables like the one above, finding the sum of BC and CA in one table and the sum of AB and CA in the other. esolutions Manual - Powered by Cognero Page 12

13 d. ALGEBRAIC Write an inequality for each of the tables you created relating the measure of the sum of two of the sides to the measure of the third side of a triangle. e. VERBAL Make a conjecture about the relationship between the measure of the sum of two sides of a triangle and the measure of the third side. a. b. It might be easier to use centimeters to measure the sides of your triangles and round to the nearest tenth place, as shown in the table below. Sample answer: c. It might be easier to use centimeters to measure the sides of your triangles and round to the nearest tenth place, as shown in the table below. Sample answer: esolutions Manual - Powered by Cognero Page 13

14 d. Compare the last two columns in each table and write an inequality comparing each of them. AB + BC > CA, BC + CA > AB, AB + CA > BC e. Sample answer: The sum of the measures of two sides of a triangle is always greater than the measure of the third side of the triangle. 43. WRITING IN MATH Analyze the information given in the diagram and explain why the markings must be incorrect. Sample answer: R is an exterior angle to, so, by the Exterior Angle Inequality, m R must be greater than m Q, one of its corresponding remote interior angles. The markings indicate that, indicating that m R = m Q. This is a contradiction of the Exterior Angle Inequality Theorem since can't be both equal to and greater than at the same time. Therefore, the markings are incorrect. esolutions Manual - Powered by Cognero Page 14

15 44. CHALLENGE Using only a ruler, draw such that m A > m B > m C. Justify your drawing. Due to the given information, that, we know that this is a triangle with three different angle measures. Therefore, it also has three different side lengths. You might want to start your construction with the longest side, which would be across from the largest angle measure. Since A is the largest angle, the side opposite it,, is the longest side. Then, make a short side across from the smallest angle measure. Since C is the smallest angle, is the shortest side. Connect points A to C and verify that the length of is between the other two sides. 45. OPEN ENDED Give a possible measure for in shown. Explain your reasoning. Since m C > m B, we know, according to Theorem 5.10, that the side opposite must be greater than the side opposite. Therefore, if AB > AC, then AB > 6 so you need to choose a length for AB that is greater than 6. Sample answer: 10; m C > m B so AB > AC, Therefore, Theorem 5.10 is satisfied since 10 > REASONING Is the base of an isosceles triangle sometimes, always, or never the longest side of the triangle? Explain. To reason through this answer, see if you can sketch isosceles triangles that have a base shorter than the two congruent legs, as well as longer than the two congruent legs. Sometimes; Sample answer: If the measures of the base angles are less than 60 degrees, then the base will be the longest leg. If the measures of the base angles are greater than 60 degrees, then the base will be the shortest leg. esolutions Manual - Powered by Cognero Page 15

16 47. CHALLENGE Use the side lengths in the figure to list the numbered angles in order from smallest to largest given that m 2 = m 5. Explain your reasoning. Walk through these angles one side at a time. Given: m 2 = m 5 The side opposite 5 is the smallest side in that triangle and m 2 = m 5, so we know that m 4 and m 6 are both greater than m 2 and m 5. Also, the side opposite m 6 is greater than the side opposite m 4. So far, we have (m 2 = m 5) < m 4 < m 6. Since the side opposite 2 is greater than the side opposite 1, we know that m 1 is less than m 2 and m 5. Now, we have m 1 < (m 2 = m 5) < m 4 < m 6. From the triangles, m 1 + m 2 + m 3 = 180 and m 4 + m 5 + m 6 = 180. Since m 2 = m 5, m 1 + m 3 must equal m 4 + m 6. Since m 1 is less than m 4, we know that m 3 is must be greater than m 6. The side lengths list in order from smallest to largest is :m 1, m 2 = m 5, m 4, m 6, m WRITING IN MATH Explain why the hypotenuse of a right triangle is always the longest side of the triangle. Based on the Triangle Sum Theorem, we know that the sum of all the angles of a triangle add up to 180 degrees. If one of the angles is 90 degrees, then the sum of the other two angles must equal degrees. If two angles add up to 90 degrees, and neither is 0 degrees, then they each must equal less than 90 degrees, making the sides across from these two acute angles shorter than the side across from the 90 degree angle, according to Theorem Because the hypotenuse is across from the 90 degree angle, the largest angle, then it must be the longest side of the triangle. esolutions Manual - Powered by Cognero Page 16

17 49. STATISTICS The chart shows the number and types of DVDs sold at three stores. According to the information in the chart, which of these statements is true? A The mean number of DVDs sold per store was 56. B Store 1 sold twice as many action and horror films as store 3 sold of science fiction. C Store 2 sold fewer comedy and science fiction than store 3 sold. D The mean number of science fiction DVDs sold per store was 46. Mean of science fiction DVDs is or 46. The correct choice is D. 50. Two angles of a triangle have measures and. What type of triangle is it? F obtuse scalene G obtuse isosceles H acute scalene J acute isosceles By Triangle Sum Theorem, the measure of the third angle is or Since no angle measures are equal, it is a scalene triangle. And one of the angles is obtuse; so, it is an Obtuse Scalene Triangle. The correct choice is F. esolutions Manual - Powered by Cognero Page 17

18 51. EXTENDED RESPONSE At a five-star restaurant, a waiter earns a total of t dollars for working h hours in which he receives \$198 in tips and makes \$2.50 per hour. a. Write an equation to represent the total amount of money the waiter earns. b. If the waiter earned a total of \$213, how many hours did he work? c. If the waiter earned \$150 in tips and worked for 12 hours, what is the total amount of money he earned? a. The equation that represents the total amount of money the waiter earns is: b. Substitute t = 213 in the equation and find h. So, the waiter worked for 6 hours. c. The total amount he earned is \$ SAT/ACT Which expression has the least value? A B C D E 99 = 99 has the least value. So, the correct choice is E. esolutions Manual - Powered by Cognero Page 18

19 In, P is the centroid, KP = 3, and XJ = 8. Find each length. 53. XK Here,. By Centroid Theorem, 54. YJ Since J is the midpoint of XY, esolutions Manual - Powered by Cognero Page 19

20 COORDINATE GEOMETRY Write an equation in slope-intercept form for the perpendicular bisector of the segment with the given endpoints. Justify your answer. 55. D( 2, 4) and E(3, 5) The slope of the segment DE is or So, the slope of the perpendicular bisector is 5. The perpendicular bisector passes through the mid point of the segment DE. The midpoint of DE is or The slope-intercept form for the equation of the perpendicular bisector of DE is: esolutions Manual - Powered by Cognero Page 20

21 56. D( 2, 4) and E(2, 1) The slope of the segment DE is or So, the slope of the perpendicular bisector is The perpendicular bisector passes through the mid point of the segment DE. The midpoint of DE is or The slope-intercept form for the equation of the perpendicular bisector of DE is: esolutions Manual - Powered by Cognero Page 21

22 57. JETS The United States Navy Flight Demonstration Squadron, the Blue Angels, flies in a formation that can be viewed as two triangles with a common side. Write a two-column proof to prove that if T is the midpoint of and. The given information in this proof can help make congruent corresponding sides for the two triangles. If T is the midpoint of, then it would form two congruent segments, and. Using this, along with the other given and the Reflexive Property for the third pair of sides, you can prove these triangles are congruent using SSS. Given: T is the midpoint of Prove: Proof: Statements(Reasons) 1. T is the midpoint of (Given) 2. (Definition of midpoint) 3. (Given) 4. (Reflexive Property) 5. (SSS) 58. POOLS A rectangular pool is 20 feet by 30 feet. The depth of the pool is 60 inches, but the depth of the water is of the depth of the pool. Find each measure to the nearest tenth. (Lesson 1-7) a. the surface area of the pool b. the volume of water in the pool 12 inches = 1 foot. Therefore, 60 inches = 5 feet. a. Surface area of the pool S is: b. The depth of the water is or 3.75ft. The volume of the water in the pool is: esolutions Manual - Powered by Cognero Page 22

23 Determine whether each equation is true or false if x = 8, y = 2, and z = The equation is false for these values The equation is false for these values. The equation is true for these values. esolutions Manual - Powered by Cognero Page 23

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