Inequalities in Two Triangles LESSON 5–6

Lesson Menu Five-Minute Check (over Lesson 5–5) TEKS Then/Now Theorems: Inequalities in Two Triangles Example 1: Use the Hinge Theorem and its Converse Proof: Hinge Theorem Example 2: Real-World Example: Use the Hinge Theorem Example 3: Apply Algebra to the Relationships in Triangles Example 4: Prove Triangle Relationships Using Hinge Theorem Example 5: Prove Relationships Using Converse of Hinge Theorem

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.12 ≤ MN ≤ 19 B.12 < MN < 19 C.5 < MN < 12 D.7 < MN < 12 Write an inequality to describe the length of MN. ___

TEKS Targeted TEKS G.6(D) Verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems. Mathematical Processes G.1(E), G.1(G)

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.

Concept

Example 2 Use the Hinge Theorem HEALTH Doctors use a straight-leg-raising test to determine the amount of pain felt in a person’s back. The patient lies flat on the examining table, and the doctor raises each leg until the patient experiences pain in the back area. Nitan can tolerate the doctor raising his right leg 35° and his left leg 65° from the table. Which leg can Nitan raise higher above the table? AnalyzeUsing the angles given in the problem, you need to determine which leg can be risen higher above the table.

Example 2 Use the Hinge Theorem FormulateDraw a diagram of the situation. DetermineSince Nitan’s legs are the same length and his left leg and the table is the same length in both situations, the Hinge Theorem says his left leg can be risen higher, since 65° > 35°.

Example 2 Use the Hinge Theorem Answer: Nitan can raise his left leg higher above the table. JustifyNitan’s left leg is pointed 30° more towards the ceiling, so it should be higher that his right leg. EvaluateThe Hinge Theorem is an effective method for comparing two triangles with congruent sides.

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

Example 4 Prove Triangle Relationships Using Hinge Theorem Write a two-column proof. Given: JK = HL m  JKH + m  HKL < m  JHK + m  KHL Prove:JH < KL StatementsReasons 1. JK = HL1. Given 2. HK = HK2. Reflexive Property 3. m  JKH + m  HKL < m  JHK + m  KHL, JH || KL 3. Given

Example 4 Prove Triangle Relationships Using Hinge Theorem StatementsReasons 5. m  JKH + m  JHK < m  JHK + m  KHL 5. Substitution 6. m  JKH < m  KHL 6. Subtraction Property of Inequality 4. m  HKL = m  JHK 4. Alternate Interior angles are 7. JH < KL7. Hinge Theorem

Which reason correctly completes the following proof? Given: Prove: AC > DC Example 4

StatementsReasons 2.2. Reflexive Property 3. m  ABC = m  ABD + m  DBC 3. Angle Addition Postulate 1.1. Given 5. AC > DC5. ? 4. m  ABC > m  DBC 4. Definition of Inequality

Example 4 A.Substitution B.Isosceles Triangle Theorem C.Hinge Theorem D.none of the above

Example 5 Prove Relationships Using Converse of Hinge Theorem Given: Prove: SP > ST

Example 5 Prove Relationships Using Converse of Hinge Theorem Proof: StatementsReasons 1.1. Given 2.2. Reflexive Property 3.3. Given 4.4. Given 5.5. Substitution Converse of the Hinge Theorem Answer:

Example 5 Which reason correctly completes the following proof? Given:X is the midpoint of ΔMCX is isosceles. CB > CM Prove:

Example 5 StatementsReasons 4.CB > CM4. Given 5.m  CXB > m  CXM 5. ? 1.X is the midpoint of MB; ΔMCX is isosceles 1. Given 2.2. Definition of midpoint 3.3. Reflexive Property Definition of isosceles triangle Isosceles Triangle Theorem 8.m  CXB > m  CMX 8. Substitution

Example 5 A.Converse of Hinge Theorem B.Definition of Inequality C.Substitution D.none of the above

Inequalities in Two Triangles LESSON 5–6