1. 5.5 Inequalities in One Triangle

2. Objectives: Students will analyze triangle measurements to decide which side is longest & which angle is largest; students will then apply the Triangle Inequality. Why? So you can find possible distances, as seen in Ex. 39 Mastery is 80% or better on 5-minute checks and practice problems.

3. Skill Develop 1: Comparing Measurements of a Triangle You may discover a relationship between the positions of the longest and shortest sides of a triangle and the position of its angles. The diagrams illustrate Theorems 5.10 and 5.11.

4. Skill Develop 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

5. Skill Develop 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.

6. Skill Develop 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. Guided Practice 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. Quick Write---1 minute What have you discovered about the relationship between and angles and side lengths in a Triangle? Explain.

9. Skill Develop 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.

10. 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

11. Theorem 5.12-Exterior Angle Inequality-------Why? 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

12. Think…..Ink….Share DIRECTOR’S CHAIR. In the director’s chair shown, AB ≅ AC and BC > AB. What can you conclude about the angles in ∆ABC?  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°.

13. 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.

14. Skill Develop 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. Pay attention to the two smallest measures then compare it to the 3 rd.

15. Skill Develop 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

16. Skill Develop 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

17. Think….Ink…Pair Share Solve the inequality: AB + AC > BC. (x + 2) +(x + 3) > 3x – 2 2x + 5 > 3x – 2 5 > x – 2 7 > x

18. Exit Slips 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.

19. What was the Objective for today? Students will analyze triangle measurements to decide which side is longest & which angle is largest; students will then apply the Triangle Inequality. Why? So you can find possible distances, as seen in Ex. 39 Mastery is 80% or better on 5-minute checks and practice problems.

20. Homework Page 331 # 2 and 6-29