5.5 Inequalities in One Triangle

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

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

Assignment pp #1-25, 34

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. 5.10 and 5.11.

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

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

Ex. 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°

Ex. 1: 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 5 7

Paragraph Proof – Theorem 5.10
Given►AC > 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.

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

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

Ex. 2: Using Theorem 5.10 DIRECTOR’S CHAIR. In the director’s chair shown, AB ≅ AC and BC > AB. What can you conclude about the angles in ∆ABC?

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°.

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.

Ex. 3: Constructing a Triangle
2 cm, 2 cm, 5 cm 3 cm, 2 cm, 5 cm 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.

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

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

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 > 4 > x 24 > x ►So, the length of the third side must be greater than 4 cm and less than 24 cm.

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

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