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ESSENTIAL QUESTION: How to use triangle measurements to decide which side is longest and which angle is largest? Theorem 4.10 If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. Example 1 Order Angle Measures Name the angles from largest to smallest. SOLUTION TV > TU, so mU > mV. Also, TU > UV, so mV > mT. ANSWER The order of the angles from largest to smallest is U, V, T. Checkpoint Order Angle Measures and Side Lengths Name the angles from largest to smallest. 1. 3. 2. Checkpoint Order Angle Measures and Side Lengths Name the angles from largest to smallest. 1. ANSWER N; L; M ANSWER Q; R; P ANSWER U; S; T 2. 3. Theorem 4.11 If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. Example 2 Order Side Lengths Name the sides from longest to shortest. SOLUTION mE > mD, so DF > FE. Also, mD > mF, so FE > DE. ANSWER The order of the sides from longest to shortest is DF, FE, DE. Checkpoint Order Angle Measures and Side Lengths Name the sides from longest to shortest. 4. 6. 5. Checkpoint Order Angle Measures and Side Lengths Name the sides from longest to shortest. 4. ANSWER GH; JG; JH ANSWER DE; EF; DF ANSWER AC; AB; BC 5. 6. Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Example 3 Use the Triangle Inequality Can the side lengths form a triangle? Explain. a. 3, 5, 9 SOLUTION a. These lengths do not form a triangle, because 3 + 5 < 9. b. 3, 5, 8 c. 3, 5, 7 b. c. These lengths do not form a triangle, because 3 + 5 = 8. These lengths do form a triangle, because 3 + 5 > 7, 3 + 7 > 5, and 5 + 7 > 3. Checkpoint Use the Triangle Inequality Can the side lengths form a triangle? Explain. 7. 5, 7, 13 8. 6, 9, 12 9. 10, 15, 25 Checkpoint Use the Triangle Inequality Can the side lengths form a triangle? Explain. 7. 5, 7, 13 ANSWER No; 5 + 7 < 13. 8. 6, 9, 12 ANSWER Yes; 6 + 9 > 12, 6 + 12 > 9, and 9 + 12 > 6. 9. 10, 15, 25 ANSWER No; 10 + 15 = 25. 1. BD is a median of ∆ABC. Find the length of AD. ANSWER 7 2. Point P is the centroid of ∆LMN and QN = 45. Find PN and QP. ANSWER PN = 30, QP = 15 3. Point D is the centroid of ∆ABC and DE = 14. Find CD and CE. ANSWER CD = 28, CE = 42 Hw Practice 4.7A