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Warm-Up : EOC Review

Warm-Up : EOC Review . What is the negation of x ≤ 10? A) x ≤ 10 B) –x ≤ 10 C) –x > 10 D) x > 10. What is the inverse of p  q? A) q  p B) ~q  ~p C) p  q D) ~p  ~q. Inequalities in Triangles 5.5. Today’s Goals By the end of class today, YOU should be able to….

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Warm-Up : EOC Review

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  1. Warm-Up:EOC Review What is the negation of x ≤ 10? A) x ≤ 10 B) –x ≤ 10 C) –x > 10 D) x > 10 What is the inverse of p  q? A) q  p B) ~q  ~p C) p  q D) ~p  ~q

  2. Inequalities in Triangles5.5 Today’s GoalsBy the end of class today, YOU should be able to… 1. Use inequalities involving angles of triangles. 2. To use inequalities involving sides of triangles.

  3. Comparison Property of Inequality If a = b + c and c > 0, then a > b.

  4. Proof of the Comparison Property of Inequality Given: a = b + c, and c > 0 Prove: a > b Statement 1: c > 0 Given Statement 2: b + c > b + 0 Addition Property of Inequality Statement 3: b + c > b Simplify Statement 4: a = b + c Given Statement 5: a > b Substitute a for b + c in Statement 3

  5. Corollary to the Triangle Exterior Angle Theorem • The measure of an exterior angle of a triangle is greater than the measure of each of its remote interior angles. • m<1 > m<2 and m<1 > m<3

  6. Theorem 5-10 • If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side. • If XZ > XY, then m<Y > m<Z.

  7. Theorem 5-11 • If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle. If m<A > m<B, then BC > AC.

  8. Ex.1: Using Theorem 5-11 • In TUV, which side is shortest?

  9. Ex.1: Solution By the Triangle Angle-Sum Theorem, m<T = 60. The smallest angle in TUV is U. It follows, by Theorem 5-11, that the shortest side is TV.

  10. You Try… • Which side is shortest? • Which side is longest?

  11. Triangle Inequality Theorem • The sum of the lengths of any two sides of a triangle is greater than the length of the third side. • XY + YZ > XZ • YZ + ZX > YX • ZX + XY > ZY

  12. Ex.2: Using the Triangle Inequality Theorem Can a triangle have sides with the given lengths? 3ft, 7ft, 8ft

  13. Ex.2: Solution 3 + 7 > 8 8 + 7 > 3 3 + 8 > 7 Yes, the sum of any two lengths is greater than the third length

  14. You Try… Can a triangle have sides with the given lengths? 3ft, 6 ft, 10 ft

  15. Ex.3: Using the Triangle Inequality Theorem A triangle has sides of lengths 8 cm and 10 cm. Describe the lengths possible for the third side.

  16. Ex.3: Solution Let x represent the length of the third side. x + 8 > 10 x > 2 x + 10 > 8 x > -2 8 + 10 > x x < 18 The third side must be longer than 2 cm and shorter than 18 cm.

  17. You Try… A triangle has sides of lengths 7 in and 4 in. Describe the lengths possible for the third side.

  18. Homework • Page 276 #s 1, 6, 7, 14, 17, 20, 24, 25 • Page 277 # 34 • The assignment can also be found at: • http://www.pearsonsuccessnet.com/snpapp/iText/products/0-13-037878-X/Ch05/05-05/PH_Geom_ch05-05_Ex.pdf

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