1. Inequalities in One Triangle Geometry

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

3. Objective 1: Comparing Measurements of a Triangle In diagrams here, you may discover a relationship between the positions of the longest and shortest sides of a triangle and the position of its angles..

4. 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. 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.

6. 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 °

7. 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

8. 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

9. 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?

10. 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 5.10. By substitution, m  A > m  B. In addition, you can conclude that m  A >60 °, m  B< 60°, and m  C < 60°.

11. 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.

12. 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

13. 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

14. 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 10 + 14 > x 24 > x ►So, the length of the third side must be greater than 4 cm and less than 24 cm.

15. Ex. 5: Solve the inequality: AB + AC > BC. (x + 2) +(x + 3) > 3x – 2 2x + 5 > 3x – 2 5 > x – 2 7 > x