5.6 and 5.7 Triangle Inequalities You found the relationship between the angle measures of a triangle. Recognize and apply properties of inequalities to the measures of the angles of a triangle. Recognize and apply properties of inequalities to the relationships between the angles and sides of a triangle. Apply the Hinge Theorem or its converse to make comparisons in two triangles. Then/Now

Concept

List the angles of ΔABC in order from smallest to largest. Order Triangle Angle Measures Example 1 List the angles of ΔABC in order from smallest to largest. Answer: C, A, B Example 2

List the angles of ΔTVX in order from smallest to largest. CYP Ex. 1 List the angles of ΔTVX in order from smallest to largest. A. X, T, V B. X, V, T C. V, T, X D. T, V, X Example 2

List the sides of ΔABC in order from shortest to longest. Order Triangle Side Lengths Example 2 List the sides of ΔABC in order from shortest to longest. Answer: AC, AB, BC Example 3

List the sides of ΔRST in order from shortest to longest. CYP Ex. 2 List the sides of ΔRST in order from shortest to longest. A. RS, RT, ST B. RT, RS, ST C. ST, RS, RT D. RS, ST, RT Example 3

Concept

Identify Possible Triangles Given Side Lengths Example 3 A. Is it possible to form a triangle with side lengths of 6 , 6 , and 14 ? If not, explain why not. __ 1 2 Check each inequality.  X Answer: Example 1

Identify Possible Triangles Given Side Lengths Example 3 B. Is it possible to form a triangle with side lengths of 6.8, 7.2, 5.1? If not, explain why not. Check each inequality. 6.8 + 7.2 > 5.1 7.2 + 5.1 > 6.8 6.8 + 5.1 > 7.2 14 > 5.1 12.3 > 6.8  11.9 > 7.2  Since the sum of all pairs of side lengths are greater than the third side length, sides with lengths 6.8, 7.2, and 5.1 will form a triangle. Answer: yes Example 1

CYP Ex. 3 B. Is it possible to form a triangle given the side lengths 4.8, 12.2, and 15.1? A. yes B. no Example 1

In ΔPQR, PQ = 7.2 and QR = 5.2. Which measure cannot be PR? A 7 B 9 Find Possible Side Lengths Example 4 In ΔPQR, PQ = 7.2 and QR = 5.2. Which measure cannot be PR? A 7 B 9 C 11 D 13 Example 2

In ΔXYZ, XY = 6, and YZ = 9. Which measure cannot be XZ? CYP Ex. 4 In ΔXYZ, XY = 6, and YZ = 9. Which measure cannot be XZ? A. 4 B. 9 C. 12 D. 16 Example 2

Concept

A. Compare the measures AD and BD. Use the Hinge Theorem and Its Converse Example 5 A. Compare the measures AD and BD. In ΔACD and ΔBCD, AC  BC, CD  CD, and mACD > mBCD. Answer: By the Hinge Theorem, mACD > mBCD, so AD > DB. Example 1

B. Compare the measures mABD and mBDC. Use the Hinge Theorem and Its Converse Example 5 B. Compare the measures mABD and mBDC. In ΔABD and ΔBCD, AB  CD, BD  BD, and AD > BC. Answer: By the Converse of the Hinge Theorem, mABD > mBDC. Example 1

A. Compare the lengths of FG and GH. CYP Ex. 5 A. Compare the lengths of FG and GH. A. FG > GH B. FG < GH C. FG = GH D. not enough information Example 1

B. Compare mJKM and mKML. CYP Ex. 5 B. Compare mJKM and mKML. A. mJKM > mKML B. mJKM < mKML C. mJKM = mKML D. not enough information Example 1

ALGEBRA Find the range of possible values for a. Apply Algebra to the Relationships in Triangles Example 6 ALGEBRA Find the range of possible values for a. From the diagram we know that Example 3

Converse of the Hinge Theorem Apply Algebra to the Relationships in Triangles Example 6 Converse of the Hinge Theorem Substitution Subtract 15 from each side. Divide each side by 9. Recall that the measure of any angle is always greater than 0. Subtract 15 from each side. Divide each side by 9. Example 3

The two inequalities can be written as the compound inequality Apply Algebra to the Relationships in Triangles Example 6 The two inequalities can be written as the compound inequality Example 3

Find the range of possible values of n. CYP Ex. 6 Find the range of possible values of n. Example 3