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Inequalities Involving Two Triangles The following two theorems are useful in determining relationships between sides and angles in triangles. Hinge Thereorm (SAS Inequality Theorem) Converse of Hinge Theorem (SSS Inequality) If two sides of one triangle are congruent to two sides of another triangle, and the included angles are not congruent, then the longer third side is opposite the larger included angle. If two sides of one triangle are congruent to two sides of another triangle, and the third sides are not congruent, then the larger included angle is opposite the longer third side. Examples: Refer to each figure to write an inequality relating the given pair of angle or segment measures. 1. 1 ____ 2 2. AD _____ BE 1 2 25 30o D 9 8 A U 3. UT ______ TS 12 R 5. 1 ____ 2 12 o 25 35o 50 N L 12 65o 1 7 65o 2 5 8 S 1 K 6. KPE ____ GPH 16 20 (3x-18)o 20 E P 12 Write an inequality or pair of inequalities to describe the possible values of x. 7. 8. 5x - 7 5 70o 70o H 100o 110o B 6 6 16 28 E o P 4. LPM ____ MPN T 22o 21o 6 50 C 10 12 42o 12 11 6 G 9. 10. 15 40 41 10 71 36 38 12. 14 26 103 50 15 5x-6 11. 3x+14 32 10 5 8 14 250° 65 65 2x+17 (13x-100)° 8 6 13. Find the shortest and longest segment in ∆DEG. D 50° 14. Find the shortest and longest segment in ∆EFG. 15. Find the shortest and longest segment in the figure. 70° 60° G E 60° 40° 80° F