D AC BC AB BD m ACB m BCD. g. Look for a pattern of the measures in your table. Then write a conjecture that summarizes your observations.

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1 OMMON O Learning tandard HG-O Inequalities in Two Triangles ssential Question If two sides of one triangle are congruent to two sides of another triangle, what can you say about the third sides of the triangles? omparing Measures in Triangles Work with a partner. Use dynamic geometry software. a. raw, as shown below. b. raw the circle with center (3, 3) through the point (, 3). c. raw so that is a point on the circle ample oints (, 3) (3, 0) (3, 3) (4.75,.03) egments = 3 = = = 3.6 =.68 d. Which two sides of are congruent to two sides of? Justify your answer. e. ompare the lengths of and. Then compare the measures of and. re the results what you expected? xplain. f. rag point to several locations on the circle. t each location, repeat part (e). opy and record your results in the table below. ONTUTING VIL GUMNT To be proficient in math, you need to make conjectures and build a logical progression of statements to explore the truth of your conjectures. m m. (4.75,.03) g. Look for a pattern of the measures in your table. Then write a conjecture that summarizes your observations. ommunicate Your nswer. If two sides of one triangle are congruent to two sides of another triangle, what can you say about the third sides of the triangles? 3. xplain how you can use the hinge shown at the left to model the concept described in Question. ection 6.6 Inequalities in Two Triangles 343

2 6.6 Lesson What You Will Learn ore Vocabulary revious indirect proof inequality ompare measures in triangles. olve real-life problems using the Hinge Theorem. omparing Measures in Triangles Imagine a gate between fence posts and that has hinges at and swings open at. s the gate swings open, you can think of, with side formed by the gate itself, side representing the distance between the fence posts, and side representing the opening between post and the outer edge of the gate. Notice that as the gate opens wider, both the measure of and the distance increase. This suggests the Hinge Theorem. Theorems Theorem 6. Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second. roof igideasmath.com V 88 X W 35 WX > T T Theorem 6.3 onverse of the Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is 9 larger than the included angle of the second. F roof xample 3, p. 345 m > m F 4 in. T 3 in. Using the onverse of the Hinge Theorem Given that T, how does m T compare to m? OLUTION You are given that T, and you know that by the eflexive roperty of ongruence (Theorem.). ecause 4 inches > 3 inches, T >. o, two sides of T are congruent to two sides of and the third side of T is longer. y the onverse of the Hinge Theorem, m T > m. 344 hapter 6 elationships Within Triangles

3 Using the Hinge Theorem Given that JK LK, how does JM compare to LM? OLUTION You are given that JK LK, and you know that KM KM by the eflexive roperty of J 64 6 M ongruence (Theorem.). ecause 64 > 6, m JKM > m LKM. o, two sides of JKM are congruent to two sides of LKM, and the included angle in JKM is larger. y the Hinge Theorem, JM > LM. K L Monitoring rogress Use the diagram. Help in nglish and panish at igideasmath.com. If = and m Q > m Q, which is longer, Q or Q?. If = and Q < Q, which is larger, Q or Q? roving the onverse of the Hinge Theorem Q Write an indirect proof of the onverse of the Hinge Theorem. Given, F, > F rove m > m Indirect roof tep ssume temporarily that m m. Then it follows that either m < m or m = m. tep If m < m, then < F by the Hinge Theorem. If m = m, then. o, F by the ongruence Theorem (Theorem 5.5) and = F. tep 3 oth conclusions contradict the given statement that > F. o, the temporary assumption that m m cannot be true. This proves that m > m. F roving Triangle elationships Y Write a paragraph proof. X Given XWY XYW, WZ > YZ rove m WXZ > m YXZ aragraph roof ecause XWY XYW, XY XW W Z by the onverse of the ase ngles Theorem (Theorem 5.7). y the eflexive roperty of ongruence (Theorem.), XZ XZ. ecause WZ > YZ, m WXZ > m YXZ by the onverse of the Hinge Theorem. Monitoring rogress Help in nglish and panish at igideasmath.com 3. Write a temporary assumption you can make to prove the Hinge Theorem indirectly. What two cases does that assumption lead to? ection 6.6 Inequalities in Two Triangles 345

4 olving eal-life roblems olving a eal-life roblem Two groups of bikers leave the same camp heading in opposite directions. ach group travels miles, then changes direction and travels. miles. Group starts due east and then turns 45 toward north. Group starts due west and then turns 30 toward south. Which group is farther from camp? xplain your reasoning. OLUTION. Understand the roblem You know the distances and directions that the groups of bikers travel. You need to determine which group is farther from camp. You can interpret a turn of 45 toward north, as shown. north east turn 45 toward north. Make a lan raw a diagram that represents the situation and mark the given measures. The distances that the groups bike and the distances back to camp form two triangles. The triangles have two congruent side lengths of miles and. miles. Include the third side of each triangle in the diagram. Group 30. mi mi amp mi. mi 45 Group 3. olve the roblem Use linear pairs to find the included angles for the paths that the groups take. Group : = 35 Group : = 50 The included angles are 35 and 50. Group. mi 50 mi amp mi 35. mi Group ecause 50 > 35, the distance Group is from camp is greater than the distance Group is from camp by the Hinge Theorem. o, Group is farther from camp. 4. Look ack ecause the included angle for Group is 5 less than the included angle for Group, you can reason that Group would be closer to camp than Group. o, Group is farther from camp. Monitoring rogress Help in nglish and panish at igideasmath.com 4. WHT IF? In xample 5, Group leaves camp and travels miles due north, then turns 40 toward east and travels. miles. ompare the distances from camp for all three groups. 346 hapter 6 elationships Within Triangles

5 6.6 xercises ynamic olutions available at igideasmath.com Vocabulary and ore oncept heck. WITING xplain why Theorem 6. is named the Hinge Theorem.. OMLT TH NTN In and F,, F, and < F. o m > m by the onverse of the Hinge Theorem (Theorem 6.3). Monitoring rogress and Modeling with Mathematics In xercises 3 6, copy and complete the statement with <, >, or =. xplain your reasoning. (ee xample.) 3. m m 4. m m OOF In xercises and, write a proof. (ee xample 4.). Given XY YZ, m WYZ > m WYX rove WZ > WX W Z 5. m m 6. m m 3 In xercises 7 0, copy and complete the statement with <, >, or =. xplain your reasoning. (ee xample.) 5. Given, < X rove m > m Y MN LK 36 3 K M 4 9. T U 0. Q U 4 T J 3 N L In xercises 3 and 4, you and your friend leave on different flights from the same airport. etermine which flight is farther from the airport. xplain your reasoning. (ee xample 5.) 3. Your flight: Flies 00 miles due west, then turns 0 toward north and flies 50 miles. Friend s flight: Flies 00 miles due north, then turns 30 toward east and flies 50 miles. 4. Your flight: Flies 0 miles due south, then turns 70 toward west and flies 80 miles. Friend s flight: Flies 80 miles due north, then turns 50 toward east and flies 0 miles. ection 6.6 Inequalities in Two Triangles 347

6 5. O NLYI escribe and correct the error in using the Hinge Theorem (Theorem 6.) Q y the Hinge Theorem (Thm. 6.), Q <. 6. T ONING Which is a possible measure for JKM? elect all that apply. K WING ONLUION The path from to F is longer than the path from to. The path from G to is the same length as the path from G to F. What can you conclude about the angles of the paths? xplain your reasoning. 8. TT ONING In FG, the bisector of F intersects the bisector of G at point H. xplain why FG must be longer than FH or HG. 9. TT ONING N is a median of NQ, and NQ > N. xplain why NQ is obtuse. G Maintaining Mathematical roficiency Find the value of x. (ection 5. and ection 5.4) 5. 7 L 8 M 6 J F MTHMTIL ONNTION In xercises 0 and, write and solve an inequality for the possible values of x x x + 3 4x 3 x. HOW O YOU IT? In the diagram, triangles are formed by the locations of the players on the basketball court. The dashed lines represent the possible paths of the basketball as the players pass. How does m compare with m? ft 8 ft ft 6 ft 6 ft 3. ITIL THINKING In, the altitudes from and meet at point, and m > m. What is true about? Justify your answer. 4. THOUGHT OVOKING The postulates and theorems in this book represent uclidean geometry. In spherical geometry, all points are on the surface of a sphere. line is a circle on the sphere whose diameter is equal to the diameter of the sphere. In spherical geometry, state an inequality involving the sum of the angles of a triangle. Find a formula for the area of a triangle in spherical geometry. eviewing what you learned in previous grades and lessons x x x 64 x 348 hapter 6 elationships Within Triangles