LESSON 5-5 INEQUALITIES IN TRIANGLES OBJECTIVE: To use inequalities involving angles and sides of triangles

Theorem 5-10 If a triangle is scalene, then the largest angle lies opposite the longest side and the smallest angle lies opposite the shortest side. 17” X Y Z 29” 32” Example 1: List the angles from smallest to largest ZZ YY XX

Theorem 5-11(Converse of Theorem 5-10) If a triangle is scalene, then the longest side lies opposite the largest angle, and the shortest side lies opposite the smallest angle. R Q S 30° Example 2: In  QRS, list the sides from smallest to largest SRQS QR

Example 3: In  TUV, which side is the shortest? 58° U V 62° Use  sum to find m  T. m  T = 60°, so  U is smallest Therefore VT is shortest T

Theorem 5-12 The Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Example 4: Can a triangle have sides with the given lengths? Explain. a) 3ft., 7ft., 8ft. b) 3cm., 6cm., 10cm. Yes, > 8NO, < 10

Example 5: A triangle has sides of lengths 8cm and 10cm. Describe the lengths possible for the third side. Let x = the length of the 3rd side. The sum of any 2 sides must be greater than the 3rd.

x + 8 > 10 x > 2 x + 10 > 8 x > > x 18 > x x < 18 So, x must be longer than 2cm & shorter than 18cm. So, there are 3 possibilities. 2 < x < 18

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