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5.5 Use Inequalities in A Triangle

5.5 Use Inequalities in A Triangle. Objectives:. Use triangle measurements to decide which side is longest or which angle is largest . Use the Triangle Inequality Theorem. largest angle. longest side. shortest side. smallest angle. Comparing Measurements of a .

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5.5 Use Inequalities in A Triangle

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  1. 5.5 Use Inequalities in A Triangle

  2. Objectives: • Use triangle measurements to decide which side is longest or which angle is largest. • Use the Triangle Inequality Theorem

  3. largest angle longest side shortest side smallest angle Comparing Measurements of a  • The longest side and largest angle of a  are opposite each other. • The shortest side and smallest angle of a  are opposite each other.

  4. 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C

  5. 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 oppositethe smaller angle. 60° 40° EF > DF

  6. Example 1: Writing Measurements in Order from Least to Greatest Write the measurements of the triangles from least to greatest. m G < mH < m J JH < JG < GH 100° 45° 35°

  7. Example 2: Writing Measurements in Order from Least to Greatest Write the measurements of the triangles from least to greatest. QP < PR < QR m R < mQ < m P 8 7 5

  8. Using the Triangle Inequality • Not every group of three segments can be used to form a triangle. The lengths of the segments must fit a certain relationship.

  9. Activity: Constructing a Triangle • 2 cm, 2 cm, 5 cm • 3 cm, 2 cm, 5 cm • 4 cm, 2 cm, 5 cm Activity: Let’s try drawing triangles with the given side lengths.

  10. Activity: Constructing a Triangle • 2 cm, 2 cm, 5 cm • 3 cm, 2 cm, 5 cm • 4 cm, 2 cm, 5 cm Notice, only group (c) is possible. Thus, what we can deduce is that the sum of the first and second lengths must be greater than the third length.

  11. Theorem 5.12: Triangle Inequality Theorem • The sum of the lengths of any two sides of a triangle is greater than the length of the third side. AB + BC > AC AC + BC > AB AB + AC > BC

  12. Example 3: Finding Possible Side Lengths If x was the smallest side, then x + 10 > 14, so x > 4 If x was the longest side, then 10 + 14 > x, so 24 > x ► So, the length of the third side must be greater than 4 cm and less than 24 cm. • A triangle has one side of 10 cm and another of 14 cm. Describe the possible lengths of the third side • SOLUTION: Let x represent the length of the third side. Using the Triangle Inequality, you can write and solve inequalities.

  13. Example 4: Using Algebra to Find Possible Side Lengths • Solve the inequality: AB + AC > BC. (x + 2) +(x + 3) > 3x – 2 2x + 5 > 3x – 2 5 > x – 2 7 > x

  14. Assignment • Workbooks Pg. 97 – 99 #4 – 9, 13 – 29

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