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Section5.6 Inequalities in One Triangle

Section5.6 Inequalities in One Triangle. Objectives: To use inequalities involving angles and sides of triangles To use inequalities involving angles and sides of triangles MA. 912.G.4.7. Corollary to the Triangle Exterior Angle Theorem.

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Section5.6 Inequalities in One Triangle

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  1. Section5.6Inequalities in One Triangle Objectives: To use inequalities involving angles and sides of triangles To use inequalities involving angles and sides of triangles MA. 912.G.4.7

  2. Corollary to the Triangle Exterior Angle Theorem The measure of an exterior angle of a triangle is greater than the measure of each of its remote interior angle. IF: 1 is an exterior angle THEN: > and > 3 2 1 We learned in chapter 3 that angles 2 and 3 must add up to be the measure of angle 1.

  3. 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

  4. 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. FE > DF

  5. Example # 1 Inequalities Within a Triangle The longest side is The largest angle is So, the largest angle is So, the longest side is

  6. Example # 2: How to write the measurements. • What these two theorems mean, is that you can put the measurements of a triangle in order, either from least to greatest, or greatest to least. Example: Write the measures of this triangle in order, from least to greatest. Angles: Sides: 80˚ Angle E, Angle D, Angle F DF, FE, DE

  7. 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. A AB + BC > AC AC + BC > AB AB + AC > BC B C

  8. Steps to determine side length • To determine if three side lengths will form a • triangle, add the lengths of the shortest two sides. • If the sum is more than the length of the third side • then the three lengths will form a triangle. • If the sum is less than or equal to the length of the third side, then the three lengths will not form a triangle.

  9. Example #3 • Can a triangle have sides with the given lengths? Explain 8m, 10m, 19m No, because 8+10=18 which is not greater than 19. 2. Can a triangle have sides with the given lengths? Explain 2 yds, 9 yds, 10 yds Yes, because 2+9=11 which is greater than 10.

  10. You Try:Can a triangle have sides with the given lengths? Explain A. 5, 6, 8 B. 5, 3, 8 C. 6, 3, 5 D. 3, 6, 8 E. 5, 6, 11 F. 5, 11, 8 • Yes • No • Yes • Yes • No • Yes

  11. Finding the range of the third side: Since the third side cannot be larger than the other two added together, we find the maximum value by adding the two sides. Since the third side and the smallest side cannot be larger than the other side, we find the minimum value by subtracting the two sides.

  12. Example 4: Given a triangle with sides of lengths 3 and 8, find the range ofpossible values for the third side. The maximum value is: 8+3>x 11>x The minimum value is: 8-3<x 5<x Range of the third side is: 5<x<11

  13. You Try: • Given a triangle with sides of length 5 and 16, find the range of possible values for the third side. 11 < x < 21 2. Given a triangle with sides of length 6 and 6, find the range ofpossible values for the third side 0 < x < 12

  14. That’s a wrap….. • Is there anything that you are still have trouble with in Chapter 5? • Is there anything that you need more help with?

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