Assignment P : 1, 2, even, 16-20, even, 27, 30, 34, 35, 36, 46, 49 Challenge Problems

Example 1 In your notebook: On a number line, graph the following inequalities: 1. x > − 5 2. − 5 ≤ x ≤ 5 3. x 5

Example 2 In your notebook graph the inequality y < 2 x – 5.

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 = If, then.

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 don’t lick the envelopes. Or eat, squash, or mix up the sticks. 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 sticks. 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 “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:

5.5 Triangle Inequalities Objectives: 1.To complete and use the Triangle Inequality and Side-Angle Inequality Theorems

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 NO YES NO

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. 5 and 20 1.Add the 2 numbers – 25 2.Subtract the 2 numbers – 15 3.Make an inequality – 12<x<25

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 and and and 30 6<x<48 6<x16 0<x<60 OBVIOUSLY, an angle cannot have a measure of 0 x<60

Example 6 Find all possible values of x. x<15

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 < 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?

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. Missing angle = 57 0 Greatest side = c Middle side = b Shortest side = a

Example 8 Prove the Side-Angle Inequality Theorem.

Example 8 Prove the Side-Angle Inequality Theorem.

Assignment P : 1, 2, even, 16-20, even, 27, 30, 34, 35, 36, 46, 49 Challenge Problems