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Aim: What is Triangle Inequality?

Aim: What is Triangle Inequality?. Do Now: Sketch 9. 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. XY + YZ > XZ. YZ + XZ > XY. XZ + XY > YZ. X. Z. Triangle Inequality Theorem.

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Aim: What is Triangle Inequality?

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  1. Aim: What is Triangle Inequality? Do Now: Sketch 9

  2. 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 XY + YZ > XZ YZ + XZ > XY XZ + XY > YZ X Z

  3. 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 X Z XY – YZ < XZ < YZ + XY

  4. TRIANGLE FORMED? NO YES YES YES YES YES NO

  5. Model Problems 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

  6. Model Problems 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. 6 cm, 6 cm 0 < z < 12 8 ft, 12 ft. 4 < z < 20

  7. Model Problems A, B, OR C? Which is the largest angle of ABC? Which is the smallest? A 7 5 A B C C 9 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

  8. Z X Y Who’s Bigger? Y Fill in the blanks x z X Z y Angles: m___ < m___ < m___ z x y Segments: ___ < ___ < ___ Conjecture: 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.

  9. Who’s Bigger? List the angles of each triangle in order from smallest to largest. S < T < R B < C < A

  10. Model Problems List the sides of each triangle in order from shortest to longest. 75o 60o 45o HJ < JK < HK BC < AC < AB

  11. What Type of Triangle? A triangle has sides measuring 3, 5, and 7. Is it a right triangle? If not what type? The Pythagorean Theorem, a2 + b2 = c2 will show us if it’s a right triangle. 7 would be the measure of the hypotenuse. a = 3, b = 5, c = 7 ? Not a right  32 + 52 = 72  9 + 25 = 49 Obtuse or acute? 32 + 52 = 72  34 < 49 a2 + b2 < c2 Obtuse  a2 + b2 > c2 Acute 

  12. What Type of Triangle? The numbers represent the length of the sides of a triangle. Classify each as acute, obtuse, or right. 13, 84, 85 132 + 842 = 852 rt. triangle 169 + 7056= 7225 7225= 7225 6, 11, 14 62 + 112 = 142 obtuse triangle 36 + 121= 196 157< 196 acute triangle 11 + 7= 16 18> 16 a2 + b2 < c2 Obtuse  a2 + b2 > c2 Acute 

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