1. 5.5 – Use Inequalities in a Triangle

2. MN P Measure each side of the triangle in centimeters and each angle in degrees. Write these measurements on your triangle. 2.6cm 5cm 5.9cm 55º30º 95º

3. Theorem: If one ______ of the triangle is longer than the other side, then the angle ____________ the __________ side is _________ than the angle ____________ the ____________ side side opposite longer larger oppositeshorter Theorem: If one _________ of a triangle is __________ than another angle, then the side opposite the _____________ angle is ___________ than the side opposite the smaller angle. angle larger longer

4. Sides: _________________ Angles: ________________ 1. List the sides and angles in order from smallest to largest.  C,  A, BB

5. Sides: _________________ Angles: ________________  R,  S, 1. List the sides and angles in order from smallest to largest. TT

6. Sides: _________________ Angles: ________________ 55° 1. List the sides and angles in order from smallest to largest.  H,  J, KK

7. Sides: _________________ Angles: ________________ 33° 1. List the sides and angles in order from smallest to largest.  E,  D, FF

8. Construct a triangle with sides of 2, 2, and 6 cm. 2cm 6cm Doesn’t work! Why?

9. 6cm 2cm Too short! 2cm What if I tried 3, 3, and 6cm?

10. 6cm 3cm Still doesn’t work!

11. Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. a b c a + b > c a + c > b b + c > a

12. 2. Is it possible to construct a triangle with the given side lengths? If not, explain why not. a. 3, 4, 5 3 + 4 > 5 3 + 5 > 4 4 + 5 > 3 Yes

13. 2. Is it possible to construct a triangle with the given side lengths? If not, explain why not. b. 1, 4, 6 1 + 4 > 6 5 > 6 No

14. 2. Is it possible to construct a triangle with the given side lengths? If not, explain why not. c. 17, 17, 33 17 + 17 > 33 17 + 33 > 17 Yes

15. 2. Is it possible to construct a triangle with the given side lengths? If not, explain why not. d. 2.2, 2.6, 4.8 2.2 + 2.6 > 4.8 4.8 > 4.8 No

16. 2. Is it possible to construct a triangle with the given side lengths? If not, explain why not. e. 6 + 10.4 > 12 6 + 12 > 10.4 10.4 + 12 > 6 Yes

17. 2. Is it possible to construct a triangle with the given side lengths? If not, explain why not. f. 7, 52, 45 7 + 52 > 45 7 + 45 > 52 52 > 52 No

18. x + 9 > 21x + 21 > 921 + 9 > x x > 12 30 > x or x < 30 x > -12 12 < x < 30 3. Describe the possible lengths of the third side of the triangle given the lengths of the other two sides. a.

19. Trick! Subtract the two sides to find the lowest and add to find the highest.

20. 3. Describe the possible lengths of the third side of the triangle given the lengths of the other two sides. 12 – 6 < 6in < x < 18in x <12 + 6 12in, 6in b.

21. 3. Describe the possible lengths of the third side of the triangle given the lengths of the other two sides. 8 – 3 < 5ft < x < 11ft x <8 + 3 3ft, 8ft c.

22. 3. Describe the possible lengths of the third side of the triangle given the lengths of the other two sides. 17 – 12 < 5cm < x < 29cm x <17 + 12 12cm, 17cm d.

23. 3. Describe the possible lengths of the third side of the triangle given the lengths of the other two sides. 13 – 7 < 6yd < x < 20yd x <13 + 7 7yd, 13yd e.

24. 3. Describe the possible lengths of the third side of the triangle given the lengths of the other two sides. 9 – 9 < 0m < x < 18m x <9 + 9 9m, 9m f.

25. 4. Solve the inequality AB + AC > BC for x. 5 + 7 > 2x – 2 12 > 2x – 2 14 > 2x 7 > x x < 7

26. 4. Solve the inequality AB + AC > BC for x. 3x + 1 + 4x – 2 > 5x + 7 7x – 1 > 5x + 7 2x – 1 > 7 2x > 8 x > 4