Triangle Inequality Theorem Students will be able to

Triangle Inequality Theorem Students will be able to apply the triangle inequality theorem to find missing angles. T. 2. G. 2: Investigate the measures of segments to determine the existence of triangles (triangle inequality theorem)

Theorems: Angle-Side Relationships in Triangles • If two sides of a triangle are not congruent, then the larger angle is opposite the longer side. Conclusion: Hypothesis: • m∠C > m∠A • AB > BC FHS Unit E 2

Theorems: Angle-Side Relationships in Triangles • If two angles of a triangle are not congruent, then the longer side is opposite the larger angle. Conclusion: Hypothesis: • AB > BC • m∠C > m∠A FHS Unit E 3

The Triangle Inequality Theorem • The sum of any two of the sides of a triangle is greater than the third side. • AB + BC > AC, • BC + AC > AB, • AC + AB > BC • Examples: Can these three measures be the sides of a triangle? Yes – 4 ft. 12 ft. and 9 ft. No, because 9+5<15 – 9 ft. 5 ft. and 15 ft. FHS Unit E 4

Shortcut to Using Triangle Inequality Theorem Tell whether a triangle can have sides with the lengths of 8, 13, and 21. No. Explain. We need to test these numbers using the Triangle Inequality Theorem, Add the smallest two numbers together and see if the sum is larger than the third number. If the sum is larger, then they can make a triangle. If the sum is not larger, then they cannot make a triangle. FHS Unit E 5

Range of Values for the Third Side • The length of two sides of a triangle are (AC ) 5 cm and (AB ) 8 cm. Find the range of possible lengths for the third side (BC). – In order to make a triangle, x must be greater than 3. x > 3 Why? A – In order to make a triangle, x 8 must be less than 13. x < 13 Why? 5 C x FHS B – Combine these inequalities to: Unit E 3 < x < 13 6

Range of Values for the Third Side • In other words, this is what we do to get to the answer. – Subtract the two given sides: 8 – 5 = 3 – Add the two given sides: 8 + 5 = 13 A 8 5 C FHS x – Plug these two numbers into the inequality: 3 < x < 13 B Unit E 7

Lesson Quiz: Part I 1. Write the angles in order from smallest to largest. C, B, A 2. Write the sides in order from shortest to longest. FHS Unit E 8

Lesson Quiz: Part II 3. The lengths of two sides of a triangle are 17 cm and 12 cm. Find the range of possible lengths for the third side. 5 cm < x < 29 cm 4. Tell whether a triangle can have sides with lengths 2. 7, 3. 5, and 9. 8. Explain. No; 2. 7 + 3. 5 > 9. 8. 5. Ray wants to place a chair so it is 10 ft from his television set. Can the other two distances shown be 8 ft and 6 ft? Explain. FHS Yes; the sum of any two lengths is greater than Unit the third length. E 9