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5-6. Inequalities in Two Triangles. Warm Up. Lesson Presentation. Lesson Quiz. Holt Geometry. Warm Up 1. Write the angles in order from smallest to largest. 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.

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## 5-6

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**5-6**Inequalities in Two Triangles Warm Up Lesson Presentation Lesson Quiz Holt Geometry**Warm Up**1.Write the angles in order from smallest to largest. 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. X, Z, Y 3 cm < s < 21 cm**Objective**Apply inequalities in two triangles.**Example 1A: Using the Hinge Theorem and Its Converse**Compare mBACand mDAC. Compare the side lengths in ∆ABC and ∆ADC. AB = AD AC = AC BC > DC By the Converse of the Hinge Theorem, mBAC > mDAC.**Example 1B: Using the Hinge Theorem and Its Converse**Compare EF and FG. Compare the sides and angles in ∆EFH angles in ∆GFH. mGHF = 180° – 82° = 98° EH = GH FH = FH mEHF > mGHF By the Hinge Theorem, EF < GF.**Example 1C: 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, mMLN > mPLN. 5k – 12 < 38 Substitute the given values. k < 10 Add 12 to both sides and divide by 5.**Example 1C Continued**Step 2 Since PLN is in a triangle, mPLN > 0°. 5k – 12 > 0 Substitute the given values. k < 2.4 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.**Check It Out! Example 1a**Compare mEGHand mEGF. Compare the side lengths in ∆EGH and ∆EGF. FG = HGEG = EGEF > EH By the Converse of the Hinge Theorem, mEGH< mEGF.**Check It Out! Example 1b**Compare BC and AB. Compare the side lengths in ∆ABD and ∆CBD. AD = DC BD = BD mADB > mBDC. By the Hinge Theorem, BC > AB.**Example 2: Travel Application**John and Luke leave school at the same time. John rides his bike 3 blocks west and then 4 blocks north. Luke rides 4 blocks east and then 3 blocks at a bearing of N 10º E. Who is farther from school? Explain.**Example 2 Continued**The distances of 3 blocks and 4 blocks are the same in both triangles. The angle formed by John’s route (90º) is smaller than the angle formed by Luke’s route (100º). So Luke is farther from school than John by the Hinge Theorem.**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.**Example 3: Proving Triangle Relationships**Write a two-column proof. Given: Prove: AB > CB Proof: 1. Given 2. Reflex. Prop. of 3. Hinge Thm.**Check It Out! Example 3a**Write a two-column proof. Given: C is the midpoint of BD. m1 = m2 m3 > m4 Prove: AB > ED**Proof:**1. Given 1.C is the mdpt. of BD m3 > m4, m1 = m2 2. Def. of Midpoint 3.1 2 3. Def. of s 4. Conv. of Isoc. ∆ Thm. 5.AB > ED 5. Hinge Thm.**Check It Out! Example 3b**Write a two-column proof. Given: SRT STR TU > RU Prove: mTSU > mRSU 1. Given 1.SRT STR TU > RU 2. Conv. of Isoc. Δ Thm. 3. Reflex. Prop. of 4. mTSU > mRSU 4. Conv. of Hinge Thm.**Lesson Quiz: Part I**1. Compare mABCand mDEF. 2. Compare PS and QR. mABC> mDEF PS < QR**Lesson Quiz: Part II**3. Find the range of values for z. –3 < z < 7**1. Given**2. Reflex. Prop. of 3. Conv. of Hinge Thm. 3. mXYW <mZWY Lesson Quiz: Part III 4. Write a two-column proof. Prove: mXYW <mZWY Given: Proof:

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