5 6 Inequalities in Two Triangles Warm Up

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5 -6 Inequalities in Two Triangles Warm Up 1. Write the angles in order

5 -6 Inequalities in Two Triangles Warm Up 1. Write the angles in order from smallest to largest. X, Z, Y 2. The lengths of two sides of a triangle are 12 cm and 9 cm. Find the range of possible lengths for the third side. 3 cm < s < 21 cm Holt Geometry

5 -6 Inequalities in Two Triangles Holt Geometry

5 -6 Inequalities in Two Triangles Holt Geometry

5 -6 Inequalities in Two Triangles Holt Geometry

5 -6 Inequalities in Two Triangles Holt Geometry

5 -6 Inequalities in Two Triangles Example 1 A: Using the Hinge Theorem and

5 -6 Inequalities in Two Triangles Example 1 A: Using the Hinge Theorem and Its Converse Compare m BAC and m DAC. Compare the side lengths in ∆ABC and ∆ADC. AB = AD AC = AC BC > DC By the Converse of the Hinge Theorem, m BAC > m DAC. Holt Geometry

5 -6 Inequalities in Two Triangles Example 1 B: Using the Hinge Theorem and

5 -6 Inequalities in Two Triangles Example 1 B: Using the Hinge Theorem and Its Converse Compare EF and FG. Compare the sides and angles in ∆EFH angles in ∆GFH. m GHF = 180° – 82° = 98° EH = GH FH = FH m EHF > m GHF By the Hinge Theorem, EF < GF. Holt Geometry

5 -6 Inequalities in Two Triangles Example 1 C: Using the Hinge Theorem and

5 -6 Inequalities in Two Triangles Example 1 C: Using the Hinge Theorem and Its Converse Find the range of values for k. Step 1 Compare the side lengths in ∆MLN and ∆PLN. LN = LN LM = LP MN > PN By the Converse of the Hinge Theorem, m MLN > m PLN. 5 k – 12 < 38 k < 10 Holt Geometry Substitute the given values. Add 12 to both sides and divide by 5.

5 -6 Inequalities in Two Triangles Example 1 C Continued Step 2 Since PLN

5 -6 Inequalities in Two Triangles Example 1 C Continued Step 2 Since PLN is in a triangle, m PLN > 0°. 5 k – 12 > 0 k < 2. 4 Substitute the given values. Add 12 to both sides and divide by 5. Step 3 Combine the two inequalities. The range of values for k is 2. 4 < k < 10. Holt Geometry

5 -6 Inequalities in Two Triangles Check It Out! Example 2 When the swing

5 -6 Inequalities in Two Triangles Check It Out! Example 2 When the swing ride is at full speed, the chairs are farthest from the base of the swing tower. What can you conclude about the angles of the swings at full speed versus low speed? Explain. The of the swing at full speed is greater than the at low speed because the length of the triangle on the opposite side is the greatest at full swing. Holt Geometry

5 -6 Inequalities in Two Triangles Example 3: Proving Triangle Relationships Write a two-column

5 -6 Inequalities in Two Triangles Example 3: Proving Triangle Relationships Write a two-column proof. Given: Prove: AD > CB Proof: Statements Reasons 1. Given 2. Reflex. Prop. of 3. Hinge Thm. Holt Geometry

5 -6 Inequalities in Two Triangles Check It Out! Example 3 a Write a

5 -6 Inequalities in Two Triangles Check It Out! Example 3 a Write a two-column proof. Given: C is the midpoint of BD. m 1 = m 2 m 3 > m 4 Prove: AB > ED Holt Geometry

5 -6 Inequalities in Two Triangles Proof: Statements 1. C is the mdpt. of

5 -6 Inequalities in Two Triangles Proof: Statements 1. C is the mdpt. of BD m 3 > m 4, m 1 = m 2 Reasons 1. Given 2. Def. of Midpoint 3. 1 2 3. Def. of s 4. Conv. of Isoc. ∆ Thm. 5. AB > ED Holt Geometry 5. Hinge Thm.

5 -6 Inequalities in Two Triangles Warm Up(Add to HW) 1. Compare m ABC

5 -6 Inequalities in Two Triangles Warm Up(Add to HW) 1. Compare m ABC and m DEF. m ABC > m DEF 2. Compare PS and QR. PS < QR Holt Geometry

5 -6 Inequalities in Two Triangles Lesson Quiz: Part II 3. Find the range

5 -6 Inequalities in Two Triangles Lesson Quiz: Part II 3. Find the range of values for z. – 3 < z < 7 Holt Geometry

5 -6 Inequalities in Two Triangles Lesson Quiz: Part III 4. Write a two-column

5 -6 Inequalities in Two Triangles Lesson Quiz: Part III 4. Write a two-column proof. Prove: m XYW < m ZWY Given: Proof: Statements Reasons 1. Given 2. Reflex. Prop. of 3. m XYW < m ZWY Holt Geometry 3. Conv. of Hinge Thm.