Concept

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

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

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

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

Concept

Concept

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

Plan Draw a diagram of the situation. 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°. Example 2

Answer: Nitan can raise his left leg higher above the table. 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. Example 2

Meena and Rita are both flying kites in a field near their houses 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 B A. Meena’s kite B. Rita’s kite Example 2

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

Converse of the Hinge Theorem 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

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

A B C D Find the range of possible values of n. 1 A. 6 < n < 12 C. n > 6 D. 6 < n < 18.3 __ 1 7 A B C D Example 3

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

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

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

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

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

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

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

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