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

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Cecilia Merritt
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the Hinge Theorem Example 3: Apply Algebra to the Relationships in Triangles Example 4: Prove

1 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 1

2 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 10. (4-5) Find the range for the measure of the third side if two sides measure 4 and 13. if two sides measure 8.3 and Write an inequality to describe the length of MN. Over Lesson 5 5 Determine whether it is possible to form a triangle with side lengths 5, 7, and 8. A. yes B. no 2

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

4 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 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 <

5 Over Lesson 5 5 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 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. 5

6 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. 6

7 Use the Hinge Theorem and Its Converse A. Compare the measures AD and BD. Notice: Also,: > Answer: By the Hinge Theorem, m ACD > m BCD, so AD > DB. Use the Hinge Theorem and Its Converse B. Compare the measures m ABD and m BDC. Notice: Also,: > Answer: By the Converse of the Hinge Theorem, m ABD > m BDC. 7

8 A. Compare the lengths of FG and GH. A. FG > GH B. FG < GH C. FG = GH D. not enough information 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 8

9 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? The Hinge Theorem says his left leg can be risen higher, since 65 > 35. 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 9

10 Apply Algebra to the Relationships in Triangles ALGEBRA Find the range of possible values for a. Notice: Also,: > Recall: >0 Find the range of possible values of n. A. 6 < n < 25 B. C. n > 6 D. 6 < n <

11 Prove Triangle Relationships Using Hinge Theorem 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 + 3. Given m KHL, JH KL 4. m HKL = m JHK 4. AIAT 5. m JKH + m JHK < m JHK + m KHL 6. m JKH < m KHL 7. JH < KL 5. Substitution 6. Subtraction 7. Hinge Thm Which reason correctly completes the following proof? Given: Prove: AC > DC Statements m ABC = m ABD + m DBC 4. m ABC > m DBC 5. AC > DC Reasons 1. Given 2. Reflexive Property 3. Angle Addition Postulate 4. Definition of Inequality 5. Hinge? Theorem 11

12 Given: Prove Relationships Using Converse of Hinge Theorem Prove: Answer: Proof: Statements Reasons Given Reflexive Property Given Given Substitution SSS Inequality 12

13 Which reason correctly completes the following proof? Given: X is the midpoint of MCX is isosceles. CB > CM Prove: Statements 1. X is the midpoint of MB; MCX is isosceles Reasons 1. Given Definition of midpoint Reflexive Property 4. CB > CM 4. Given 5. m CXB > m CXM 5.? Converse of Hinge Thm Definition of isosceles triangle Isosceles Triangle Theorem 8. m CXB > m CMX 8. Substitution 13

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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:

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