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Exploring Triangle Inequalities: Theorems and Applications

In this section, we delve into the fundamental theorems of triangle inequalities. Theorem 5.10 posits that if one side of a triangle is longer than another, the angle opposite the longer side is also larger. Conversely, Theorem 5.11 states that if an angle is larger, the side opposite is longer. Moreover, we discuss the Exterior Angle Inequality, where an exterior angle exceeds either non-adjacent interior angle, and the Triangle Inequality, emphasizing that the sum of any two sides must be greater than the third. Engage with exercises to strengthen understanding.

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Exploring Triangle Inequalities: Theorems and Applications

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  1. Section 5.5 Inequalities in One Triangle

  2. Theorem 5.10 • If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. 15 B 12 A

  3. THEOREM 5.11 • If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle 60 o Side 2 Side 1 > Side 2 45 o Side 1

  4. Write the sides/angles in order from least to greatest. A E 15 33 63 C 32 B F 22 D

  5. Is PQ>8? Is RQ<8? Q 57 61 P R 8

  6. Exterior Angle Inequality • The measure of an exterior angle of a triangle is greater then the measure of either of the two nonadjacent interior angles. A 1 B <1 is greater than <A <1 is greater than <B

  7. What are the possible angle measures of <A? 42 A

  8. Triangle Inequality • The sum of the lengths of any two sides of a triangle is greater than the length of the third side. A AB + BC >AC AC + BC > AB AB + AC > BC C B

  9. Is it possible to have a triangle with the given side lengths? • 3, 8, 3 • 6, 7, 12 • 9, 5, 11 • 8, 12, 20

  10. What are the possible lengths of the third side of the triangle? • 8, 17, ? • 12, 18, ?

  11. Write and solve the inequality PQ + QR > PR. Q 2x+1 3x-3 P R 3x+1

  12. Homework • p. 298 #2-19, 24, 25

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