Unit 7 Lesson 1 Use Inequalities in a

Unit 7 Lesson 1 – Use Inequalities in a Triangle

P 95º 5 cm 2. 6 cm 55º M 30º 5. 9 cm N Measure each side of the triangle in centimeters and each angle in degrees. Write these measurements on your triangle.

Theorem: If one ______ side of the triangle is longer than the other side, then the angle opposite larger ______ the _____ longer side is _____ opposite shorter than the angle ______ the ______ side angle Theorem: If one _____ of a triangle is larger _____ than another angle, then the side larger opposite the _______ angle is longer ______ than the side opposite the smaller angle.

1. List the sides and angles in order from smallest to largest. Sides: _________ C, A, B Angles: ________

1. List the sides and angles in order from smallest to largest. Sides: _________ R, S, T Angles: ________

1. List the sides and angles in order from smallest to largest. Sides: _________ H, J, K Angles: ________ 55°

1. List the sides and angles in order from smallest to largest. Sides: _________ 33° E, D, F Angles: ________

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

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

3 cm 6 cm Still doesn’t work!

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+c>b b+c>a c a b

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

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

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

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 No

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 + 12 > 6 Yes

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

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

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

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

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

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

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

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

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

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