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EXAMPLE 1

Given that ST PR , how does. PST compare to SPR ?. You are given that ST PR and you know that PS PS by the Reflexive Property. Because 24 inches > 23 inches , PT >RS . So, two sides of STP are congruent to two sides of PRS and the third side in STP is longer.

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EXAMPLE 1

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  1. Given that ST PR , how does PSTcompare to SPR? You are given that ST PRand you know that PS PSby the Reflexive Property. Because 24inches>23inches,PT >RS. So, two sides of STPare congruent to two sides of PRSand the third side in STPis longer. ANSWER By the Converse of the Hinge Theorem, mPST>mSPR. EXAMPLE 1 Use the Converse of the Hinge Theorem SOLUTION

  2. BIKING Two groups of bikers leave the same camp heading in opposite directions. Each group goes 2 miles, then changes direction and goes 1.2 miles. Group A starts due east and then turns 45 toward north as shown. Group B starts due west and then turns 30 toward south. o o EXAMPLE 2 Solve a multi-step problem Which group is farther from camp? Explain your reasoning.

  3. o o Because 150 > 135 , Group B is farther from camp by the Hinge Theorem. ANSWER EXAMPLE 2 Solve a multi-step problem SOLUTION Draw a diagram and mark the given measures. The distances biked and the distances back to camp form two triangles, with congruent 2 mile sides and congruent 1.2mile sides. Add the third sides of the triangles to your diagram. Next use linear pairs to find and mark the included angles of 150°and 135°.

  4. If PR = PSand mQPR > m QPS, which is longer, SQ or RQ? 1. ANSWER RQ for Examples 1 and 2 GUIDED PRACTICE Use the diagram at the right.

  5. RPQor SPQ? If PR =PSand RQ < SQ, which is larger, 2. ANSWER SPQ for Examples 1 and 2 GUIDED PRACTICE Use the diagram at the right.

  6. WHAT IF? In Example 2, suppose Group C leaves camp and goes 2 miles due north. Then they turn 40°toward east and continue 1.2miles. Comparethe distances from camp for all three groups. for Examples 1 and 2 GUIDED PRACTICE 3. ANSWER Group B is the farthest from camp, followed by Group C, and then Group A which is the closest.

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