1. Lesson 7.4
Inequalities pp

2. Objectives: 1. To state and prove the Longer Side Inequality.
2. To state and prove the Hinge Theorem.

3. Theorem 7.12
Longer Side Inequality. One side of a triangle is longer than another side if and only if the measure of the angle opposite the longer side is greater than the measure of the angle opposite the shorter side.

4. Which angle is the largest? Which angle is the smallest?X Y Z 7 5 4
Which angle is the largest?
Which angle is the smallest?

5. Which side is the longest? Which side is the shortest?X Y Z
100°
20°
60°
Which side is the longest?
Which side is the shortest?

6. Practice: List the angles from smallest to largest. B C 6 3 5
Practice: List the angles from smallest to largest.
1. A,B,C 2. C,B,A
3. B,C,A 4. C,A,B

7. Practice: List the angles from smallest to largest.
AB = 4, BC = 7, AC = 5
1. A, B, C 2. C, B, A
3. B, C, A 4. C, A, B

8. Theorem 7.13
Hinge Theorem. Two triangles have two pairs of congruent sides. If the measure of the included angle of the first triangle is larger than the measure of the other included angle, then the opposite (3rd) side of the first triangle is longer than the opposite side of the second triangle.

9. A B C
If mA  mW, then CB  YX. W X Y

10. Homework pp

11. ►A. Exercises 1. Give the order of sides from smallest to largest. C AB
70°
65°
45°

12. ►A. Exercises 3. Give the order of sides from smallest to largest. M N
66°
50°
64°

13. ►A. Exercises 5. Give the order of angles from smallest to largest. F14 15 27

14. ►A. Exercises 7. Give the order of angles from smallest to largest. 43 5 X Y Z

15. ►A. Exercises
For each pair of triangles, compare an unlabeled pair of sides or angles. 9. U M F Y Z I 5 7 6 8

16. ►A. Exercises
For each pair of triangles, compare an unlabeled pair of sides or angles.
11. A B C 5 3 Q P R 3 5

17. Given: ∆ABC; B is right angle Prove: AC is the longest side
►A. Exercises
Prove the following statement.
13. In a right triangle the hypotenuse is the longest side.
Given: ∆ABC; B is right angle
Prove: AC is the longest side A B C

18. ►B. Exercises Prove the following statement.
14. The shortest segment from a point to a line is a perpendicular segment.

19. ►B. Exercises
14. Given: AB  BC
Prove: AC  AB A B C D E

20. ■ Cumulative Review 21. mPBA if l || m
Give an algebraic expression or a numerical value for each indicated angle.
21. mPBA if l || m P A B
37 l m

21. ■ Cumulative Review 22. mA
Give an algebraic expression or a numerical value for each indicated angle.
22. mA B A C D E

22. ■ Cumulative Review 23. mABC
Give an algebraic expression or a numerical value for each indicated angle.
23. mABC A B C x

23. ■ Cumulative Review 24. mR
Give an algebraic expression or a numerical value for each indicated angle.
24. mR Q R P x

24. ■ Cumulative Review 25. mA
Give an algebraic expression or a numerical value for each indicated angle.
25. mA B A C x
2x – 5