Inequalities in One Triangle Geometry Objectives Use triangle
Inequalities in One Triangle Geometry
Objectives: • Use triangle measurements to decide which side is longest or which angle is largest. • Use the Triangle Inequality
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. .
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
If one ANGLE of a 60° 40° triangle is larger than another ANGLE, then the SIDE opposite the larger angle is longer than the EF > DF side opposite the smaller angle. 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. a. 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. b. QP < PR < QR m R < m Q < m P 8 5 7
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 5. 10. By substitution, m A > m B. In addition, you can conclude that m A >60°, m B< 60°, and m C < 60°.
Ex. 3: Constructing a Triangle a. 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.
Ex. 3: Constructing a Triangle a. 2 cm, 5 cm b. 3 cm, 2 cm, 5 cm c. 4 cm, 2 cm, 5 cm
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
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.
Ex. 5: • Solve the inequality: AB + AC > BC. (x + 2) +(x + 3) > 3 x – 2 2 x + 5 > 3 x – 2 5>x– 2 7>x
- Slides: 15