2. Course: Applied Geometry Aim: What is Triangle Inequality Theorem Aim: What is Triangle Inequality? Do Now: Sketch 9

3. Course: Applied Geometry Aim: What is Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Y ZX XY + YZ > XZ YZ + XZ > XY XZ + XY > YZ

4. Course: Applied Geometry Aim: What is Triangle Inequality Theorem The length of the third side must be somewhere in between the SUM and the DIFFERENCE of the lengths of the first two sides. Y ZX XY – YZ < XZ < YZ + XY

5. Course: Applied Geometry Aim: What is Triangle Inequality Theorem TRIANGLE FORMED? NO YES NO YES

6. Course: Applied Geometry Aim: What is Triangle Inequality Theorem Is it possible to form a triangle with side lengths 4 cm., 6 cm., and 10 cm.? Explain No, 6 + 4 is not > 10, even though 6 - 4 = 2 < 10 Is it possible to form a triangle with side lengths 10 cm., 8 cm., and 3.5 cm.? Explain yes, 10 + 8 > 3.5 and 10 - 8 = 2 < 3.5 Model Problems

7. Course: Applied Geometry Aim: What is Triangle Inequality Theorem The lengths of two sides of a triangle are given. Write an inequality to represent the range of values for z, the length of the third side. Model Problems 8 ft, 12 ft. 4 < z < 20 6 cm, 6 cm 0 < z < 12

8. Course: Applied Geometry Aim: What is Triangle Inequality Theorem 5 7 CB A 9  A,  B, OR  C? Which is the largest angle of  ABC? Which is the smallest? AA CC If one side of a triangle is greater in measure than a second side, then the angle opposite the first side is greater in measure than the angle opposite the second side. If BC > AC, then m  A > m  B. If m  A > m  B, then BC > AC. Also true Model Problems

9. Course: Applied Geometry Aim: What is Triangle Inequality Theorem In a triangle, the angle with the smallest measure is opposite the shortest side of the triangle. The angle with the greatest measure is opposite the longest side of the triangle. Y ZX m  ___ < m  ___ < m  ___ Fill in the blanks Segments: Angles: ___ < ___ < ___ Conjecture: ZXY x z y zxy Who’s Bigger?

10. Course: Applied Geometry Aim: What is Triangle Inequality Theorem Who’s Bigger? List the angles of each triangle in order from smallest to largest.  B <  C <  A  S <  T <  R

11. Course: Applied Geometry Aim: What is Triangle Inequality Theorem Model Problems List the sides of each triangle in order from shortest to longest. HJ < JK < HK 60 o BC < AC < AB 45 o 75 o

12. Course: Applied Geometry Aim: What is Triangle Inequality Theorem A triangle has sides measuring 3, 5, and 7. Is it a right triangle? If not what type? The Pythagorean Theorem, a 2 + b 2 = c 2 will show us if it’s a right triangle. 7 would be the measure of the hypotenuse. 3 2 + 5 2 = 7 2  9 + 25 = 49 Not a right  ? a 2 + b 2 < c 2 a 2 + b 2 > c 2 Obtuse  Acute  a = 3, b = 5, c = 7 3 2 + 5 2 = 7 2  34 < 49 Obtuse or acute? What Type of Triangle?

13. Course: Applied Geometry Aim: What is Triangle Inequality Theorem The numbers represent the length of the sides of a triangle. Classify each as acute, obtuse, or right. What Type of Triangle? 13, 84, 85 a 2 + b 2 < c 2 a 2 + b 2 > c 2 Obtuse  Acute  13 2 + 84 2 = 85 2 169 + 7056 = 7225 7225 = 7225 rt. triangle 6, 11, 146 2 + 11 2 = 14 2 36 + 121 = 196 157 < 196 obtuse triangle 11 + 7 = 16 18 > 16 acute triangle