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5.5 Inequalities in One Triangle Geometry Mrs. Spitz Fall, 2004

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

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Assignment pp #1-25, 34

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Objective 1: Comparing Measurements of a Triangle In activity 5.5, you may have discovered a relationship between the positions of the longest and shortest sides of a triangle and the position of its angles. The diagrams illustrate Thms and 5.11.

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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

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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. EF > DF 60 ° 40 ° You can write the measurements of a triangle in order from least to greatest.

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

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Ex. 1: Writing Measurements in Order from Least to Greatest Write the measurements of the triangles from least to greatest. b.QP < PR < QR m R < m Q < m P 8 5 7

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Paragraph Proof – Theorem 5.10 GivenAC > AB Prove m ABC > m C Use the Ruler Postulate to locate a point D on AC such that DA = BA. Then draw the segment BD. In the isosceles triangle ABD, 1 2. Because m ABC = m 1+m 3, it follows that m ABC > m 1. Substituting m 2 for m 1 produces m ABC > m 2. Because m 2 = m 3 + m C, m 2 > m C. Finally because m ABC > m 2 and m 2 > m C, you can conclude that m ABC > m C.

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NOTE: The proof of 5.10 in the slide previous uses the fact that 2 is an exterior angle for BDC, so its measure is the sum of the measures of the two nonadjacent interior angles. Then m 2 must be greater than the measure of either nonadjacent interior angle. This result is stated in Theorem 5.12

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Theorem 5.12-Exterior Angle Inequality The measure of an exterior angle of a triangle is greater than the measure of either of the two non adjacent interior angles. m 1 > m A and m 1 > m B

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Ex. 2: Using Theorem 5.10 DIRECTORS CHAIR. In the directors chair shown, AB AC and BC > AB. What can you conclude about the angles in ABC?

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Ex. 2: Using Theorem 5.10 Solution Because AB AC, ABC is isosceles, so B C. Therefore, m B = m C. Because BC>AB, m A > m C by Theorem By substitution, m A > m B. In addition, you can conclude that m A >60 °, m B< 60°, and m C < 60°.

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Objective 2: 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.

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Ex. 3: Constructing a Triangle a.2 cm, 2 cm, 5 cm b.3 cm, 2 cm, 5 cm c.4 cm, 2 cm, 5 cm Solution: Try drawing triangles with the given side lengths. Only group (c) is possible. The sum of the first and second lengths must be greater than the third length.

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Ex. 3: Constructing a Triangle a.2 cm, 2 cm, 5 cm b.3 cm, 2 cm, 5 cm c.4 cm, 2 cm, 5 cm

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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 AC + BC > AB AB + AC > BC

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Ex. 4: Finding Possible Side Lengths 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. x + 10 > 14 x > > x 24 > x So, the length of the third side must be greater than 4 cm and less than 24 cm.

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#24 - homework Solve the inequality: AB + AC > BC. (x + 2) +(x + 3) > 3x – 2 2x + 5 > 3x – 2 5 > x – 2 7 > x

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5. Geography AB + BC > AC MC + CG > MG > x 264 > x x + 99 < 165 x < < x < 264

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