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

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? UnderstandUsing the angles given in the problem, you need to determine which leg can be risen higher above the table.

Example 2 Use the Hinge Theorem PlanDraw 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°.