5-5 5-5 Inequalities in Triangles 1. Plan GO for Help What You’ll Learn Check Skills You’ll Need • To use inequalities involving Graph the triangles with the given vertices. List the sides in order from shortest to longest. 1–4. See back of book. angles of triangles • To use inequalities involving sides of triangles . . . And Why To locate the largest corners on a triangular backyard deck, as in Example 2 Lessons 1-8 and 5-4 1. A(5, 0), B(0, 8), C(0, 0) 2. P(2, 4), Q(-5, 1), R(0, 0) 3. G(3, 0), H(4, 3), J(8, 0) 4. X(-4, 3), Y(-1, 1), Z(-1, 4) 1 2 To use inequalities involving angles of triangles To use inequalities involving sides of triangles Examples 1 2 3 4 Recall the steps for indirect proof. 5. You want to prove m&A . m&B. Assume that mlA K mlB. Write the ﬁrst step of an indirect proof. 6. In an indirect proof, you deduce that AB $ AC is false. What conclusion can you make? AB R AC 1 Objectives 5 Applying the Corollary Real-World Connection Using Theorem 5-11 Using the Triangle Inequality Theorem Finding Possible Side Lengths Math Background Inequalities Involving Angles of Triangles Theorems 5-10 and 5-11 can be treated as extending the Isosceles Triangle Theorem and its converse to the case of inequality. These theorems enable students to prove in Exercise 41 that the shortest segment from a point to a line is perpendicular to the line. When you empty a container of juice into two glasses, it is difﬁcult to be sure that the glasses get equal amounts. You can be sure, however, that each glass holds less than the original amount in the container. This is a simple application of the Comparison Property of Inequality. More Math Background: p. 256D Key Concepts Property Comparison Property of Inequality Lesson Planning and Resources If a = b + c and c . 0, then a . b. Proof See p. 256E for a list of the resources that support this lesson. Proof of the Comparison Property Given: a = b + c, c . 0 PowerPoint Prove: a . b Statements 1. c.0 2. b + c . b + 0 3. b + c . b 4. a=b+c 5. a .b Bell Ringer Practice Reasons 1. 2. 3. 4. 5. Check Skills You’ll Need Given Addition Property of Inequality Simplify. Given Substitute a for b + c in Statement 3. For intervention, direct students to: Finding Distance Lesson 1-8: Example 1 Extra Skills, Word Problems, Proof Practice, Ch. 1 The Comparison Property of Inequality allows you to prove the following corollary to the Exterior Angle Theorem for triangles (Theorem 3-13). Lesson 5-5 Inequalities in Triangles Special Needs Below Level L1 To illustrate that Theorem 4-10 and 4-11 only apply to one triangle, draw TUV along with a similar but smaller triangle, ABC. Show that TV AB does not imply that TV AC. learning style: visual 289 Indirect Proof Lesson 5-4: Examples 3, 5 Extra Skills, Word Problems, Proof Practice, Ch. 5 L2 Using geometry software to alter the sides and angles of triangles (beginning with an isosceles triangle) may help students understand Theorems 5-10 and 5-11. learning style: visual 289 2. Teach Key Concepts Corollary Corollary to the Triangle Exterior Angle Theorem The measure of an exterior angle of a triangle is greater than the measure of each of its remote interior angles. Guided Instruction 3 1 EXAMPLE Remind students that a corollary is both a statement that follows directly from a theorem and a theorem itself. 2 1 2 m&1 . m&2 and m&1 . m&3 Proof Proof of the Corollary Given: &1 is an exterior angle of the triangle. Prove: m&1 . m&2 and m&1 . m&3. Proof: By the Exterior Angle Theorem, m&1 = m&2 + m&3. Since m&2 . 0 and m&3 . 0, you can apply the Comparison Property of Inequality and conclude that m&1 . m&2 and m&1 . m&3. Tactile Learners EXAMPLE Have students construct a triangle to model the problem, using sides 18 cm, 21 cm, and 27 cm long to see that the larger angles are opposite the longer sides. 1 Applying the Corollary Additional Examples 4 X 1 3 3 P 1 By the corollary to the Exterior Angle Theorem, m&1 . m&3. So, m&2 . m&3 by substitution. 1 Explain why m&4 m&5. B 5 Y 2 Quick Check 4 2 1 Explain why m&OTY . m&3. mlOTY S ml2 T by the Comparison Prop. of Ineq. Since it was proven that ml2 S ml3, then by the Trans. Prop. mlOTY S ml3. You will prove the following inequality theorem in the exercises. Y C Key Concepts ml4 S ml2 by the Corollary to the Exterior Angle Theorem, ml2 ≠ ml5 because l2 and l5 are congruent corresponding angles, and ml4 S ml5 by substitution. Theorem 5-10 Y If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side. If XZ . XY, then m&Y . m&Z. 2 In RGY, RG = 14, GY = 12, and RY = 20. List the angles from largest to smallest. lG, lY, lR 2 Real-World Connection Careers Landscape architects blend structures with decorative plantings. Quick Check 290 EXAMPLE Real-World Z X Connection Deck Design A landscape architect is designing a triangular deck. She wants to place benches in the two larger corners. Which corners have the larger angles? Corners B and C have the larger angles. They are opposite the two longer sides of 27 ft and 21 ft. 2 List the angles of #ABC in order from smallest to largest. lA, lC, lB A 27 ft C 21 ft 18 ft B Chapter 5 Relationships Within Triangles Advanced Learners English Language Learners ELL L4 Have students find the range of possible values for the length of the third side of a triangle whose other side lengths are a and b. If bL a, b–a RcRa±b. 290 O In the diagram, m&2 = m&1 by the Isosceles Triangle Theorem. Explain why m&2 . m&3. PowerPoint A EXAMPLE learning style: verbal Use the proof of the corollary before Example 1 to review the meaning of the term corollary. A corollary is a statement that can be proved easily by applying the theorem. learning style: verbal 2 1 Guided Instruction Inequalities Involving Sides of Triangles Theorem 5-10 on the preceding page states that the larger angle is opposite the longer side. The converse is also true. Key Concepts Theorem 5-11 If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle. B C A Indirect Proof of Theorem 5-11 Given: m&A . m&B Prove: BC . AC Step 2 If BC , AC, then m&A , m&B (Theorem 5-10). This contradicts the given fact that m&A . m&B. Therefore, BC , AC must be false. If BC = AC, then m&A = m&B (Isosceles Triangle Theorem). This also contradicts m&A . m&B. Therefore, BC = AC must be false. Step 3 The assumption BC AC is false, so BC . AC. A 1 4 5 A A B B E D D C C Math Tip EXAMPLE Discuss why “x 2 and x -2” can be written as x 2. Point out that the possible lengths are written as a compound inequality. PowerPoint 3 In ABC, &C is a right angle. Which is the longest side? AB Using Theorem 5-11 E D C B EXAMPLE EXAMPLE Additional Examples E D C B A 3 C B A 2 3 E D C B Error Prevention Ask: Why does comparing only the sum of the two shorter sides and the longest side tell whether a triangle can have the given lengths? If the least sum is greater than the greatest length, the other inequalities must be true also. 5 Step 1 Assume BC AC. That is, assume BC , AC or BC = AC. EXAMPLE Emphasize that Theorems 5-10 and 5-11 apply only within triangles, not between triangles. 4 If m&A . m&B, then BC . AC. Proof 3 D E E Test-Taking Tip Don’t be distracted! Choice B lists the sides in order, but from longest to shortest, not shortest to longest. Multiple Choice Which choice shows the sides of #TUV in order from shortest to longest? TV, UV, UT UT, UV, TV UV, UT, TV T TV, UT, UV U 58 V By the Triangle Angle-Sum Theorem, m&T 60. 58 60 62, so m&U m&T m&V. By Theorem 5-11, TV UV UT. The correct choice is A. Quick Check 62 4 Can a triangle have sides with the given lengths? Explain. a. 2 cm, 2 cm, 4 cm no; 2 ± 2 4 b. 8 in., 15 in., 12 in. yes; 8 ± 12 S 15 3 List the sides of the #XYZ in order from shortest to longest. Explain your listing. YZ R XY R XZ since mlY ≠ 80. X 5 In FGH, FG = 9 m and GH = 17 m. Describe the possible lengths of FH. 8 R FH R 26 Y 40 60 Z The lengths of three segments must be related in a certain way to form a triangle. 3 cm 3 cm 2 cm Resources • Daily Notetaking Guide 5-5 L3 • Daily Notetaking Guide 5-5— L1 Adapted Instruction 2 cm 5 cm 6 cm 3 cm, 3 cm, 5 cm 2 cm, 2 cm, 6 cm Closure Explain why each triangle below is impossible. Notice that only one of the sets of three segments above can form a triangle. The sum of the smallest two lengths must be greater than the greatest length. This is Theorem 5-12 (see next page). You will prove it in the exercises. Lesson 5-5 Inequalities in Triangles 12 100° 15 50° 30° 25 291 18 12 32 In the first triangle, the side opposite the smallest angle is not the shortest side; the second triangle violates the Triangle Inequality Theorem. 291 3. Practice Key Concepts Theorem 5-12 Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Assignment Guide Y XY + YZ . XZ 1 A B 1-9, 30, 33 YZ + ZX . YX 2 A B 10-29, 31, 32, 34-37 C Challenge 38-41 Test Prep Mixed Review 42-49 50-61 4 Homework Quick Check Using the Triangle Inequality Theorem Can a triangle have sides with the given lengths? Explain. To check students’ understanding of key skills and concepts, go over Exercises 5, 22, 29, 30, 33. Exercise 5 Ask: How do you know x 0? Side lengths must be positive numbers. EXAMPLE Z X ZX + XY . ZY a. 3 ft, 7 ft, 8 ft Quick Check Visual Learners Exercises 7–9, 13–15 Have 3 + 6 10 8+7.3 The sum of 3 and 6 is not greater than 10, contradicting Theorem 5-12. The sum of any two lengths is greater than the third length. No 4 Can a triangle have sides with the given lengths? Explain. a. 2 m, 7 m, and 9 m b. 4 yd, 6 yd, and 9 yd no; 2 ± 7 w 9 yes; 4 ± 6 S 9; 6 ± 9 S 4; and 4 ± 9 S 6 5 students draw and label each triangle to make sure that they identify opposite angles correctly. 3+7.8 3 + 8 . 7 Yes For: Triangle Inequality Activity Use: Interactive Textbook, 5-5 Exercise 6 Make sure that students can explain why &I is the largest angle in GHI. b. 3 cm, 6 cm, 10 cm EXAMPLE Finding Possible Side Lengths Algebra A triangle has sides of lengths 8 cm and 10 cm. Describe the lengths possible for the third side. Let x represent the length of the third side. By the Triangle Inequality Theorem, x + 8 . 10 x + 10 . 8 x.2 8 + 10 . x x . -2 x , 18 The third side must be longer than 2 cm and shorter than 18 cm. Quick Check GPS Guided Problem Solving L3 L4 Enrichment L2 Reteaching L1 Adapted Practice Practice Name Class Practice 5-5 Inequalities in Triangles 1 ft N M C 2. 3. 25 m B Practice and Problem Solving A Practice by Example 4.1 cm 1.9 cm D 6 ft For more exercises, see Extra Skill, Word Problem, and Proof Practice. Explain why ml1 S ml2. 1–3. See margin. Q 14 m 18 m 5.5 ft Example 1 EXERCISES L3 Date Determine the two largest angles in each triangle. 1. 5 A triangle has sides of lengths 3 in. and 12 in. Describe the lengths possible for the third side. 9 R x R 15 S 4.0 cm Example 1 R 1. P 5. 13 cm I 7 cm A 15 yd (page 290) 39 cm K 20 yd 11 cm R S 6. 24 cm T 25 yd B Can a triangle have sides with the given lengths? Explain. 7. 4 m, 7 m, and 8 m 8. 6 m, 10 m, and 17 m 9. 4 in., 4 in., and 4 in. 10. 1 yd, 9 yd, and 9 yd 11. 11 m, 12 m, and 13 m 12. 18 ft, 20 ft, and 40 ft 14. 8 12 yd, 9 14 yd, and 18 yd 15. 2.5 m, 3.5 m, and 6 m 13. 1.2 cm, 2.6 cm, and 4.9 cm 3. 3 GO for Help 55 cm A 2. 2 L 4. 3 2 2 1 1 4 1 4 3 List the sides of each triangle in order from shortest to longest. B 16. C 47ⴗ 17. L S 18. 75ⴗ T © Pearson Education, Inc. All rights reserved. 41ⴗ 56ⴗ O 107ⴗ R A 292 B List the angles of each triangle in order from largest to smallest. 19. S 1.7 3.4 D 21. P S 20. Chapter 5 Relationships Within Triangles 28 R 13 25 2.6 38 26 N A J 21 O The lengths of two sides of a triangle are given. Describe the lengths possible for the third side. 22. 4 in., 7 in. 23. 9 cm, 17 cm 24. 5 ft, 5 ft 25. 11 m, 20 m 26. 6 km, 8 km 27. 24 in., 37 in. 292 1. l3 O l2 because they are vertical ' and ml1 S ml3 by Corollary to the Ext. l Thm. So, ml1 S ml2 by subst. 2. An ext. l of a k is larger than either remote int. l. 3. ml1 S ml4 by Corollary to the Ext. l Thm. and l4 O l2 because if n lines, then alt. int. ' are O. 16. No; 2 ± 3 w 6. 17. Yes; 11 ± 12 S 15; 12 ± 15 S 11; 11 ± 15 S 12. 4 Example 2 (page 290) 4. K 5. lD, lC, lE 4.3 2.7 M C 5.8 L lM, lL, lK 7. #ABC, where AB = 8, BC = 5, and CA = 7 lA, lB, lC Example 3 (page 291) Careers List the angles of each triangle in order from smallest to largest. D 6. Exercise 28 Traveling sales H 4 x 105 E 3x G 6 lG, lH, lI I 9. #XYZ,where XY = 12, YZ = 24, and ZX = 30 lZ, lX, lY List the sides of each triangle in order from shortest to longest. 10. O 8. #DEF, where DE = 15, EF = 18, and DF = 5 lE, lF, lD 11. G 12. 28 Exercise 29 Check that students T use the Triangle Inequality Theorem in their explanation. 45 (page 292) Example 5 (page 292) H 30 U N V F TU, UV, TV MN, ON, MO FH, GF, GH 13. #ABC, with 14. #DEF, with 15. #XYZ, with m&A = 90, m&D = 20, m&X = 51, m&B = 40, and m&E = 120, and m&Y = 59, and m&C = 50 m&F = 40 m&Z = 70 AC, AB, CB EF, DE, DF ZY, XZ, XY Can a triangle have sides with the given lengths? Explain. 16–21. See margin. M Example 4 110 75 16. 2 in., 3 in., 6 in. 17. 11 cm, 12 cm, 15 cm 18. 8 m, 10 m, 19 m 19. 1 cm, 15 cm, 15 cm 20. 2 yd, 9 yd, 10 yd 21. 4 m, 5 m, 9 m 27. 20 km, 35 km 15 R s R 5 28. Error Analysis The Shau family is crossing Kansas on Highway 70. A sign reads “Wichita 90 miles, Topeka 110 miles.” Avi says, “I didn’t know that it was only 20 miles from Wichita to Topeka.” Explain to Avi why the distance between the two cities doesn’t have to be 20 miles. See margin. Suppose two sides of one triangle are congruent to two sides of another triangle. If the included angle of the ﬁrst triangle is larger than the included angle of the second triangle, then 9. a. Draw a diagram to illustrate the hypothesis. b. The conclusion of the Hinge Theorem concerns the sides opposite the two angles mentioned in the hypothesis. Write the conclusion. c. Draw a diagram to illustrate the converse. d. Converse of the Hinge Theorem Write the conclusion to this theorem. Suppose two sides of one triangle are congruent to two sides of another triangle. If the third side of the ﬁrst triangle is greater than the third side of the second triangle, then 9. Lesson 5-5 Inequalities in Triangles 18. No; 8 ± 10 w 19. 19. Yes; 1 ± 15 S 15; 15 ± 15 S 1. 20. Yes; 2 ± 9 S 10; 9 ± 10 S 2; 2 ± 10 S 9. geometry software to investigate the Hinge Theorem and its converse. 23. 5 in., 16 in.11 R s R 21 24. 6 cm, 6 cm 0 R s R 12 29. Writing Explain why the distance between the two peaks in the photograph is greater than the difference of the distances from the hiker to each of the peaks. See margin. 30. The Hinge Theorem The hypothesis of the Hinge Theorem is stated below. GPS The conclusion is missing. a–d. See margin. Exercise 29 Exercise 30 Students can use possible for the third side. 25. 18 m, 23 m 5 R s R 41 26. 4 yd, 7 yd 3 R s R 11 Apply Your Skills Technology Tip x 2 Algebra The lengths of two sides of a triangle are given. Describe the lengths 22. 8 ft, 12 ft 4 R s R 20 B representatives make a great effort to plan their routes efficiently. When traveling to multiple destinations over the course of a day, week, or month, a sales representative makes a schedule of routes that minimizes travel time and maximizes time with customers. 21. No; 4 ± 5 w 9. 28. Answers may vary. Sample: If Y is the distance between Wichita and Topeka, then 20 R Y R 200. 293 29. Let the distance between the peaks be d and the distances from the hiker to each of the peaks be a and b. Then d ± a S b and d ± b S a. Thus, d S b – a and d S a – b. 30. a. D A B C E F b. The third side of the 1st k is longer than the third side of the 2nd k. c. See diagram in part (a). d. The included l of the first k is greater than the included l of the second k. 31. Answers may vary. Sample: The shortcut across the grass is shorter than the sum of the two paths. 293 4. Assess & Reteach 31. Shortcuts Explain how the student in the photograph is applying the Triangle Inequality Theorem. See margin. x 2 32. Algebra Find the longest side of #ABC, if m&A = 70, m&B = 2x - 10, and PowerPoint m&C = 3x + 20. AB Lesson Quiz 33. Developing Proof Fill in the blanks for a proof of Theorem 5-10: If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side. Use the figure for Exercises 1–3. Given: #TOY, with YO . YT. 25 2 3 4 12 3 P 1 Mark P on YO so that YP > YT. Draw TP. 1 B O Prove: a. 9 . b. 9 mlOTY; ml3 A C 27° D 1. Explain why m&4 m&1. ml4 S ml1 by the Corollary to the Exterior Angle Theorem. Real-World Statements Connection 2. Explain why m&3 m&1. By Theorem 5-10, if two sides are not congruent, the larger angle lies opposite the longer side, so ml3 S ml1. d. 9 l Add. Post. 4. m&OTY . m&2 e. 9 Comparison Prop. of Ineq. Y f. 9 Sub. (step 2) g. 9 An ext. l of a k is greater than either remote int. l. 7. m&OTY . m&3 h. 9 Trans. Prop. of Ineq. 34. Prove this corollary to Theorem 5-11: P The perpendicular segment from a point to a line is the shortest segment from the point to the line. See margin Given: PT ' TA p. 295. T Prove: PA . PT A 6. m&1 . m&3 Proof 4. Can a triangle have lengths of 2 mm, 3 mm, and 6 mm? Explain. no; 2 ± 3 6 Critical Thinking Determine which segment is shortest in each diagram. GO 6. In PQT, m&P = 50 and m&T = 70. Which side is shortest? QT Homework Help P 30 Visit: PHSchool.com Web Code: aue-0505 40 R Challenge 36. C Q 35. RS nline C 110 37. D CD 32 114 S B X 48 30 A XY W Y 95 47 40 Z 38. Probability A student has two straws, one 6 cm long and the other 9 cm long. She picks a third straw at random from a group of four straws whose lengths are 3 cm, 5 cm, 11 cm, and 15 cm. What is the probability that the straw she picks will allow her to form a triangle? 12 Alternative Assessment Have each student write a paragraph explaining the Comparison Property of Inequality, the Corollary to the Exterior Angle Theorem, the two theorems that relate the relative positions of the angles and sides of a triangle, and the Triangle Inequality Theorem. 3. m&OTY = m&4 + m&2 1. Ruler Post. 5. m&OTY . m&1 3. Use ABC to describe the possible lengths of AC . 13 R AC R 37 5. In XYZ, XY = 5, YZ = 8, and XZ = 7. Which angle is largest? lX 2. m&1 = m&2 2 T c. 9 Base ' of an isos. k are O. 1. YP > YT Some recall the Triangle Inequality Theorem as “The shortest path between two points is the straight path.” 4 Reasons 39. (2, 4), (2, 6), (3, 3), (3, 4), (3, 5), (3, 6), (3, 7), (4, 3), (4, 4), (4, 5), (4, 6), (4, 7), (4, 8) For Exercises 39 and 40, x and y are whole numbers, 1 R x R 5, and 2 R y R 9. 39. The sides of a triangle are 5 cm, x cm, and y cm. List possible (x, y) pairs. See left. 40. Probability What is the probability that you can draw an isosceles triangle that 5 has sides 5 cm, x cm, and y cm, with x and y chosen at random? 18 Proof 41. Prove Theorem 5-12: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Given: #ABC C See margin. Prove: AC + CB . AB ) A (Hint: On BC mark a point D not on BC, so that DC = AC. Draw DA and use Theorem 5-11 with #ABD.) 34. lT is the largest l in kPTA. Thus PA S PT because the longest side of a k is opp. the largest l. 294 Chapter 5 Relationships Within Triangles 41. D C A 294 B CD ≠ AC is given so kACD is isosc. by def. of isosc. k. This means mlD ≠ mlCAD. Then mlDAB S mlCAD by the Comparison Prop. of Ineq. So by subst., mlDAB S mlD and by Thm. 5-11 DB S AB. Since DC ± CB ≠ DB, by subst. DC ± CB S AB. Using subst. again, AC ± CB S AB. B Test Prep Test Prep Multiple Choice Resources P 42. For #PQR, which is the best estimate for PR? D A. 137 m B. 145 m C. 163 m D. 187 m (2a + 12) 184 m 43. Two sides of a triangle measure 13 and 15. Which length is NOT possible for the third side? F F. 2 G. 8 H. 14 J. 20 Q 44. Which statement is true for the ﬁgure at the right? D A. JN . JB B. JN . BN C. The shortest side is JB. D. The longest side is BN. B 4a 114 145 m R For additional practice with a variety of test item formats: • Standardized Test Prep, p. 301 • Test-Taking Strategies, p. 296 • Test-Taking Strategies with Transparencies J 4c (c - 25) 130 N not to scale 45. For #ABC, what must be true about an exterior angle at A? J F. It is larger than &A. G. It is smaller than &A. H. It is larger than &B. J. It is smaller than &C. 46. Which lengths can be lengths for the sides of a triangle? C A. 1, 2, 5 B. 3, 2, 5 C. 5, 2, 5 D. 7, 2, 5 47. For #JKL, L J , JK , KL. What must be true about angles J, K, and L? J F. m&L , m&J , m&K G. m&L . m&J . m&K H. m&J , m&L , m&K J. m&J . m&L . m&K Short Response 48. In #PQR, PQ . PR . QR. One angle measures 170°. List all possible whole number values for m&P. 1, 2, 3, 4, 5, 6, 7, 8, 9 49. In #ABC, m&A . m&C . m&B. a–b. See margin. a. Of AB and AC , one measures 5 inches and the other measures 9 inches. Which measures 9 inches? Explain. b. Based on your conclusion for part (a), ﬁnd all possible whole-number measures for the third side. Explain. Mixed Review GO for Help Lesson 5-4 Write the negation of each statement. 50. m&A # m&B mlA S mlB Lesson 2-5 Lesson 1-9 51. m&X . m&B mlX K mlB 52. The angle is a right angle. 53. The triangle is not obtuse. The triangle is obtuse. The angle is not a right angle. Use the diagram. Find the measure of each angle. E C 54. &ADH 90 55. &GDH 35 A 35 D 56. &CDH 145 57. &ADG 55 G H Find to the nearest tenth of a square unit the area of each circle with the given radius r or diameter d. 58. r = 1.6 ft 8.0 ft2 lesson quiz, PHSchool.com, Web Code: aua-0505 49. [2] a. Since mlA S mlC S mlB, the sides opp. them are related in the same way: BC S AB S AC. Of AB and AC , AB 59. d = 35 mm 962.1 mm2 60. r = 0.5 m 0.8 m2 61. d = 20 mi 314.2 mi 2 Lesson 5-5 Inequalities in Triangles is longer than AC . Since 9 in. S 5 in., AB ≠ 9 in. and AC ≠ 5 in. b. BC is the longest side, and 9 R BC R 14. The possible 295 whole number measures for BC are 10 in., 11 in., 12 in., and 13 in. [1] part (a) OR part (b) incorrect 295

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