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Triangle Inequalities Theorems & Examples

Understand triangle inequalities with Theorems 5.10, 5.11, and 5.12, learn about the Exterior Angle Inequality and Triangle Inequality Theorem, and discover examples showing application in triangle sides and angles relationships.

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Triangle Inequalities Theorems & Examples

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  1. Chapter 5Section 5.5 Inequalities in a triangle

  2. -3 -3 +9 -2x -10 -10 +9 -2x -1 2 -1 2 Don’t forget to change it! 3 < 2x - 9 x < 11 2 > -x 12 < 2x -2<x 6<x

  3. Compare Sides to Order Angles C B A Theorem 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. mA > mB > mC CB > CA > AB

  4. Compare Angles to Order Sides R P Q Theorem 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. mP > mQ > mR QR > PR > PQ

  5. mM > mK > mL KL > LM > KM

  6. mM > mO > mN ON > MN > OM

  7. mB > mA > mC AC > BC> AB

  8. EG > GF > EF mF > mE > mG

  9. mJ > mI > mH HI > HJ > IJ

  10. LM > KM > KL mK > mL > mM

  11. In ABC CB > AB > AC In CBD BD > CD > CB Thus BD > CD > CB > AB > AC

  12. In LKM LM > KL > KM In LMN MN > LN > LM Thus MN > LN > LM > LN > LM

  13. Inequalities in a triangle R P Q T Theorem Theorem 5.12 Exterior Angle Inequality The measure of an exterior angle of a triangle is greater than the measure of the two nonadjacent interior angles. TQR is an exterior angle of QRP mTQR > mR And mTQR > mP

  14. Inequalities in a triangle Theorem Theorem 5.13 Triangle Inequality The sum of the lengths of any two sides of a triangle is greater than the length of the third side. AB + BC > AC AB + AC > BC BC + AC > AB

  15. The sum of any two sides must be greater than the third side! YZ + XY > XZ XZ + XY > YZ XZ + YZ > XY 2 + XY > 3 3 + XY > 2 2 + 3 > XY XY > 1 5 > XY XY > -1 5 > XY > 1

  16. The sum of any two sides must be greater than the third side! YZ + XY > XZ XZ + XY > YZ XZ + YZ > XY 8 + XY > 10 10 + XY > 8 8 + 10 > XY XY > 2 18 > XY XY > -2 18 > XY > 2

  17. CC 60 AF 40 DS 60 + x > 40 60 + 40 > x x + 40 > 60 100 > x x > -20 x > 20 x 100 > x > 20 No, the distance must be less than 100 miles

  18. HW #59Pg

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