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5.5 Triangle Inequalities, Thompson Objectives 1. 2. 3. 4. To complete and use the Triangle Inequality Matching shortest to longest… Can you make a triangle… Possible range of third side Making Triangles, YAY! Given three segments of any length, can you construct a triangle? Investigation 1 So this is what happens when two sides of a “triangle” together are smaller than the third side: And here’s what happens with two sides of a “triangle” together are equal to the third side: Definition of Inequality For any real numbers a and b, a > b if and only if there is a positive number c such that a = b + c. Thus: 10 > 6 because 10 = 6 + 4 If m1 + m2 = m3, then m1 < m3. > versus ≥ Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Example Determine whether it is possible to draw a triangle with sides of the given measures. 1. 2. 3. 1 cm, 2 cm, 3 cm 21 in, 32 in, 18 in 11 m, 6 m, 2 m Look at range of possible third sides here… Example The two measures of two sides of a triangle are given. Between what two numbers must the measure of the third side fall? Write your answer as a compound inequality. 1. 21 and 27 2. 5 and 11 3. 30 and 30 Investigation 2 Use the following Investigation to discover the relationship between the measures of angles in triangles and the lengths of the sides opposite them. Investigation 2 1. 2. Draw a large scalene triangle. Guys draw acute triangles, and the Gals should draw obtuse triangles. Measure the angles in each triangle. Label the angle with greatest measure L, the angle with second greatest measure M, and the remaining angle S. Investigation 2 3. Measure the three sides. Which side is the longest? Label it by placing the lowercase letter l near the middle of the side. Which side is the second longest? Label it m in the same way. Which side is the shortest? Label it s. Investigation 2 Which side, l, m, or s, is opposite the angle with the greatest measure? Which side is opposite the angle with the least measure? Investigation 2 Which side, l, m, or s, is opposite the angle with the greatest measure? Which side is opposite the angle with the least measure? Example 7 In the triangle at the right, put the unknown measures in order from greatest to least. 55 b c 68 a Classwork Page 260 14 - 22 Page 277 5 – 27 odd 32, 34-36 Page 155 25, 54 Page 162 13, 15, 21 Page 284 Chapter Test (skip 1-6, 15)