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Inequalities in Triangles: Hinge Theorem and Converse

Learn how to use the Hinge Theorem and its converse to compare sides and angles in triangles. Apply algebra to find the range of possible values for triangle relationships. Includes real-world examples and proofs.

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Inequalities in Triangles: Hinge Theorem and Converse

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 5–5) CCSS Then/Now Theorems: Inequalities in Two Triangles Example 1: Use the Hinge Theorem and its Converse Proof: Hinge Theorem Example 2: Real-World Example: Use the Hinge Theorem Example 3: Apply Algebra to the Relationships in Triangles Example 4: Prove Triangle Relationships Using Hinge Theorem Example 5: Prove Relationships Using Converse of Hinge Theorem Lesson Menu

  3. Determine whether it is possible to form a triangle with side lengths 5, 7, and 8. A. yes B. no 5-Minute Check 1

  4. Determine whether it is possible to form a triangle with side lengths 4.2, 4.2, and 8.4. A. yes B. no 5-Minute Check 2

  5. Determine whether it is possible to form a triangle with side lengths 3, 6, and 10. A. yes B. no 5-Minute Check 3

  6. Find the range for the measure of the third side of a triangle if two sides measure 4 and 13. A. 5 < n < 12 B. 6 < n < 16 C. 8 < n < 17 D. 9 < n < 17 5-Minute Check 4

  7. Find the range for the measure of the third side of a triangle if two sides measure 8.3 and 15.6. A. 11.7 < n < 25.4 B. 9.1 < n < 22.7 C. 7.3 < n < 23.9 D. 6.3 < n < 18.4 5-Minute Check 5

  8. ___ Write an inequality to describe the length of MN. A. 12 ≤ MN ≤ 19 B. 12 < MN < 19 C. 5 < MN < 12 D. 7 < MN < 12 5-Minute Check 6

  9. Content Standards G.CO.10 Prove theorems about triangles. Mathematical Practices 3 Construct viable arguments and critique the reasoning of others. 1 Make sense of problems and persevere in solving them. CCSS

  10. You used inequalities to make comparisons in one triangle. • Apply the Hinge Theorem or its converse to make comparisons in two triangles. • Prove triangle relationships using the Hinge Theorem or its converse. Then/Now

  11. Concept

  12. In ΔACD and ΔBCD, AC BC, CD  CD, and mACD > mBCD. Use the Hinge Theorem and Its Converse A. Compare the measures AD and BD. Answer: By the Hinge Theorem, mACD > mBCD, so AD > DB. Example 1

  13. In ΔABD and ΔBCD, AB CD, BD  BD, and AD > BC. Use the Hinge Theorem and Its Converse B. Compare the measures mABD and mBDC. Answer: By the Converse of the Hinge Theorem, mABD > mBDC. Example 1

  14. A. Compare the lengths of FG and GH. A.FG > GH B.FG < GH C.FG = GH D. not enough information Example 1

  15. B. Compare mJKM and mKML. A.mJKM > mKML B.mJKM < mKML C.mJKM = mKML D. not enough information Example 1

  16. Concept

  17. Use the Hinge Theorem HEALTH Doctors use a straight-leg-raising test to determine the amount of pain felt in a person’s back. The patient lies flat on the examining table, and the doctor raises each leg until the patient experiences pain in the back area. Nitan can tolerate the doctor raising his right leg 35° and his left leg 65° from the table. Which leg can Nitan raise higher above the table? Understand Using the angles given in the problem,you need to determine which leg can be risen higher above the table. Example 2

  18. Use the Hinge Theorem Plan Draw a diagram of the situation. SolveSince Nitan’s legs are the same length and his left leg and the table is the same length in both situations, the Hinge Theorem says his left leg can be risen higher, since 65° > 35°. Example 2

  19. Use the Hinge Theorem Answer: Nitan can raise his left leg higher above the table. CheckNitan’s left leg is pointed 30° more towards the ceiling, so it should be higher that his right leg. Example 2

  20. Meena and Rita are both flying kites in a field near their houses. Both are using strings that are 10 meters long. Meena’s kite string is at an angle of 75° with the ground. Rita’s kite string is at an angle of 65° with the ground. If they are both standing at the same elevation, which kite is higher in the air? A. Meena’s kite B. Rita’s kite Example 2

  21. From the diagram we know that Apply Algebra to the Relationships in Triangles ALGEBRA Find the range of possible values for a. Example 3

  22. Apply Algebra to the Relationships in Triangles Converse of the Hinge Theorem Substitution Subtract 15 from each side. Divide each side by 9. Recall that the measure of any angle is always greater than 0. Subtract 15 from each side. Divide each side by 9. Example 3

  23. The two inequalities can be written as the compound inequality Apply Algebra to the Relationships in Triangles Example 3

  24. A.6 < n < 25 B. C.n > 6 D.6 < n < 18.3 Find the range of possible values of n. Example 3

  25. Write a two-column proof. Given: JK = HL; JH || KL mJKH + mHKL < mJHK + mKHL Prove: JH < KL Statements Reasons 1.JK = HL 1. Given 2. HK = HK 2. Reflexive Property 3. mJKH + mHKL < mJHK + mKHL, JH || KL 3. Given Prove Triangle Relationships Using Hinge Theorem Example 4

  26. Statements Reasons 4.mHKL = mJHK 4. Alternate Interior angles are 5. mJKH + mJHK < mJHK + mKHL 5. Substitution 6. mJKH< mKHL 6. Subtraction Property of Inequality 7. JH< KL 7. Hinge Theorem Prove Triangle Relationships Using Hinge Theorem Example 4

  27. Which reason correctly completes the following proof?Given:Prove: AC > DC Example 4

  28. Statements Reasons 5. AC > DC 5. ? 4.mABC > mDBC 4. Definition of Inequality 1. 1. Given 2. 2. Reflexive Property 3. mABC =mABD + mDBC 3. Angle Addition Postulate Example 4

  29. A. Substitution B. Isosceles Triangle Theorem C. Hinge Theorem D. none of the above Example 4

  30. Given: Prove: Prove Relationships Using Converse of Hinge Theorem Example 5

  31. Proof: Statements Reasons 1. 1. Given 2. 2. Reflexive Property 3. 3. Given 4. 4. Given 5. 5. Substitution 6. 6. SSS Inequality Prove Relationships Using Converse of Hinge Theorem Answer: Example 5

  32. Which reason correctly completes the following proof? Given:X is the midpoint ofΔMCX is isosceles.CB > CM Prove: Example 5

  33. Statements Reasons 1. X is the midpoint of MB; ΔMCX is isosceles 1. Given 2. 2. Definition of midpoint 3. 3. Reflexive Property 4. CB > CM 4. Given 5. mCXB > mCXM 5. ? 6. 6. Definition of isosceles triangle 7. Isosceles Triangle Theorem 7. 8. Substitution 8. mCXB >mCMX Example 5

  34. A. Converse of Hinge Theorem B. Definition of Inequality C. Substitution D. none of the above Example 5

  35. End of the Lesson

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