Homework Assignment Page 322 #3-15 Page 323 #17-22, #25-27, 29-31, 33-35

Use Inequalities in a Triangle LESSON 5.5

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 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.

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. EF > DF 60 ° 40 ° 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 45 ° 100° 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 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 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.

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

Theorem 5.12: 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 > > 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 > x 264 > x x + 99 < 165 x < < x < 264

Assignment pp #3-19, 31-34, 37-39