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5.5 Triangle Inequalities Objectives: 1. To complete and use the Triangle Inequality Theorems Assignment: β’ P. 331-334: 1, 2, 6-12 even, 16-20, 22-26 even, 27, 30, 34, 35, 36, 46, 49 β’ Challenge Problems Objective 1 You will be able to complete and use the Triangle Inequality Theorems Example 1 On a number line, graph the following inequalities: 1. π₯ > β5 2. β5 β€ π₯ β€ 5 3. π₯ < β5 or π₯ > 5 Example 2 Use a whiteboard to graph the inequality π¦ < 2π₯ β 5. Definition of Inequality For any real numbers π and π, π > π if and only if there is a positive number π such that π = π + π. 10 > 6 because 10 = 6 + 4 If πβ 1 + πβ 2 = πβ 3, then πβ 1 > πβ 3 Example 3 Given three segments of any length, can you construct a triangle? Investigation 1 Use the following investigation to complete the Triangle Inequality Theorem. Oh, and donβt lick the envelopes. Or eat, squash, or mix up the straws. Thanks. Investigation 1 1. Assemble a triangle with each set of straws. You are not allowed to cut, bend, or otherwise change the size or shape of each straw. 2. Were you able to construct a triangle each time? Why or why not? Investigation 1 So this is what happens when two sides of a βtriangleβ together are smaller than the third side: And hereβs what happens with two sides of a βtriangleβ together are equal to the third side: 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 4 Determine whether it is possible to draw a triangle with sides of the given measures. 1. 1 cm, 2 cm, 3 cm 2. 21 in, 32 in, 18 in 3. 11 m, 6 m, 2 m Example 5 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 Example 6 Find all possible values of x. 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. Draw a large scalene triangle. Some group members should draw acute triangles, and some should draw obtuse triangles. Investigation 2 2. Measure the angles in each triangle. Label the angle with greatest measure β πΏ, the angle with second greatest measure β π, and the remaining angle β π. π π π πΏ πΏ π 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. π π π π π π πΏ π πΏ π π π 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. π π π π π π πΏ π πΏ π π π 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? Side-Angle Inequality Theorem If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. Angle-Side Inequality Theorem If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. Example 7 In the triangle at the right, put the unknown measures in order from greatest to least. 55ο° b c 68ο° a Example 8 Prove the Side-Angle Inequality Theorem. A D 1 2 3 C B Example 8 Prove the Side-Angle Inequality Theorem. A D 1 2 3 C B 5.5 Triangle Inequalities Objectives: 1. To complete and use the Triangle Inequality and SideAngle Inequality Theorems Assignment: β’ P. 331-334: 1, 2, 6-12 even, 16-20, 22-26 even, 27, 30, 34, 35, 36, 46, 49 β’ Challenge Problems