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:

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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

3 Over Lesson 5 5 Determine whether it is possible to form a triangle with side lengths 5, 7, and 8. A. yes B. no

4 Over Lesson 5 5 Determine whether it is possible to form a triangle with side lengths 5, 7, and 8. A. yes B. no

5 Over Lesson 5 5 Determine whether it is possible to form a triangle with side lengths 4.2, 4.2, and 8.4. A. yes B. no

6 Over Lesson 5 5 Determine whether it is possible to form a triangle with side lengths 4.2, 4.2, and 8.4. A. yes B. no

7 Over Lesson 5 5 Determine whether it is possible to form a triangle with side lengths 3, 6, and 10. A. yes B. no

8 Over Lesson 5 5 Determine whether it is possible to form a triangle with side lengths 3, 6, and 10. A. yes B. no

9 Over Lesson 5 5 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

10 Over Lesson 5 5 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

11 Over Lesson 5 5 Find the range for the measure of the third side of a triangle if two sides measure 8.3 and A < n < 25.4 B. 9.1 < n < 22.7 C. 7.3 < n < 23.9 D. 6.3 < n < 18.4

12 Over Lesson 5 5 Find the range for the measure of the third side of a triangle if two sides measure 8.3 and A < n < 25.4 B. 9.1 < n < 22.7 C. 7.3 < n < 23.9 D. 6.3 < n < 18.4

13 A. 12 MN 19 Over Lesson 5 5 Write an inequality to describe the length of MN. B. 12 < MN < 19 C. 5 < MN < 12 D. 7 < MN < 12

14 A. 12 MN 19 Over Lesson 5 5 Write an inequality to describe the length of MN. B. 12 < MN < 19 C. 5 < MN < 12 D. 7 < MN < 12

15 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.

16 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.

17

18 Use the Hinge Theorem and Its Converse A. Compare the measures AD and BD. In ΔACD and ΔBCD, AC BC, CD CD, and m ACD > m BCD. Answer:

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

20 Use the Hinge Theorem and Its Converse B. Compare the measures m ABD and m BDC. In ΔABD and ΔBCD, AB CD, BD BD, and AD > BC. Answer:

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

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

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

24 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

25 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

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27 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.

28 Use the Hinge Theorem Plan Draw a diagram of the situation. Solve Since 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.

29 Answer: Use the Hinge Theorem

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

31 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

32 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

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

34 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.

35 Apply Algebra to the Relationships in Triangles The two inequalities can be written as the compound inequality

36 Apply Algebra to the Relationships in Triangles The two inequalities can be written as the compound inequality

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

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

39 Prove Triangle Relationships Using Hinge Theorem Write a two-column proof. Given: JK = HL; JH KL Prove: m JKH + m HKL < m JHK + m KHL 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

40 Statements Prove Triangle Relationships Using Hinge Theorem 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

41 Which reason correctly completes the following proof? Given: Prove: AC > DC

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

43 A. Substitution B. Isosceles Triangle Theorem C. Hinge Theorem D. none of the above

44 A. Substitution B. Isosceles Triangle Theorem C. Hinge Theorem D. none of the above

45 Given: Prove Relationships Using Converse of Hinge Theorem Prove:

46 Answer: Proof: Statements Prove Relationships Using Converse of Hinge Theorem Reasons Given Reflexive Property Given Given Substitution SSS Inequality

47 Which reason correctly completes the following proof? Given: X is the midpoint of ΔMCX is isosceles. CB > CM Prove:

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

49 A. Converse of Hinge Theorem B. Definition of Inequality C. Substitution D. none of the above

50 A. Converse of Hinge Theorem B. Definition of Inequality C. Substitution D. none of the above

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Over Lesson 5 5 (1-3) Determine whether it is possible to form a triangle with the given lengths of sides:. 5, 7, and , 4.2, and , 6, and 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

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