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Lesson Menu Five-Minute Check (over Lesson 5–5) 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

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

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

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

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

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

Over Lesson 5–5 A.A B.B C.C D.D 5-Minute Check 5 A.12 ≤ MN ≤ 19 B.12 < MN < 19 C.5 < MN < 12 D.5 < MN < 19 Write an inequality to describe the length of MN. ___

Then/Now You used inequalities to make comparisons in one triangle. (Lesson 5–3) Apply the Hinge Theorem or its converse to make comparisons in two triangles. Prove triangle relationships using the Hinge Theorem or its converse.

Concept

Example 1 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. In ΔACD and ΔBCD, AC  BC, CD  CD, and  ACD >  BCD.

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

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

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

Example 2 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? PlanDraw a diagram of the situation. Answer: Nitan can raise his left leg higher above the table.

A.A B.B Example 2 A.Meena’s kite B.Rita’s kite 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?

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

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

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

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

StatementsReasons 2.2. Reflexive Property 3. m  ABC > m  ABD + m  DBC 3. Given 1.1. Given 5. AC > DC5. ? 4. m  ABC > m  DBC 4. Definition of Inequality

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

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

Example 5 Prove Relationships Using Converse of Hinge Theorem Proof: StatementsReasons 1.1. Given 2.2. Reflexive Property 3.3. Given 4.4. Given 5.5. Substitution SSS Inequality

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

Example 5 StatementsReasons 4.CB > CM4. Given 5.m  CXB > m  CXM 5. ? 1.X is the midpoint of MB; ΔMCX is isosceles 1. Given 2.2. Definition of midpoint 3.3. Reflexive Property Definition of isosceles triangle Isosceles Triangle Theorem 8.m  CXB > m  CMX 8. Substitution

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

End of the Lesson