Over Lesson 5–5 5-Minute Check 1 A.yes B.no Determine whether it is possible to form a triangle with side lengths 5, 7, and 8.

Over Lesson 5–5 5-Minute Check 2 A.yes B.no Determine whether it is possible to form a triangle with side lengths 4.2, 4.2, and 8.4.

Over Lesson 5–5 5-Minute Check 3 A.yes B.no Determine whether it is possible to form a triangle with side lengths 3, 6, and 10.

Over Lesson 5–5 5-Minute Check 4 A.5 < n < 12 B.6 < n < 16 C.8 < n < 17 D.9 < n < 17 Find the range for the measure of the third side of a triangle if two sides measure 4 and 13.

Over Lesson 5–5 5-Minute Check 5 A.11.7 < n < 25.4 B.9.1 < n < 22.7 C.7.3 < n < 23.9 D.6.3 < n < 18.4 Find the range for the measure of the third side of a triangle if two sides measure 8.3 and 15.6.

Over Lesson 5–5 5-Minute Check 6 A.5 ≤ MN ≤ 19 B.5 < MN < 19 C.5 < MN < 12 D.7 < MN < 12 Write an inequality to describe the length of MN. ___

Then/Now You used inequalities to make comparisons in one triangle. Apply the Hinge Theorem or its converse to make comparisons in two triangles. Prove triangle relationships using the Hinge Theorem or its converse.

Concept

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

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

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

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

Example 2 A.Meena’s kite B.Rita’s kite Meena and Rita are both flying kites in a field near their houses. Both are using strings that are 10 meters long. Meena’s kite string is at an angle of 75° with the ground. Rita’s kite string is at an angle of 65° with the ground. If they are both standing at the same elevation, which kite is higher in the air?

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

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

Example 3 Find the range of possible values of n. A.6 < n < 25 B. C.n > 6 D.6 < n < 18.3