UNIT 5.7 INEQUALITIESUNIT 5.7 INEQUALITIESIN TWO TRIANGLESIN TWO TRIANGLES

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

Apply inequalities in two triangles.

Objective

Example 1A: Using the Hinge Theorem and Its Converse

Compare m∠BAC and m∠DAC.

Compare the side lengths in ∆ABC and ∆ADC.

By the Converse of the Hinge Theorem, m∠BAC > m∠DAC.

AB = AD AC = AC BC > DC

Example 1B: Using the Hinge Theorem and Its Converse

Compare EF and FG.

By the Hinge Theorem, EF < GF.

Compare the sides and angles in ∆EFH angles in ∆GFH.

EH = GH FH = FH m∠EHF > m∠GHF

m∠GHF = 180° – 82° = 98°

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.

By the Converse of the Hinge Theorem, m∠MLN > m∠PLN.

LN = LN LM = LP MN > PN

5k – 12 < 38

k < 10

Substitute the given values.

Add 12 to both sides and divide by 5.

Example 1C Continued

Step 2 Since ∠PLN is in a triangle, m∠PLN > 0°.

Step 3 Combine the two inequalities.

The range of values for k is 2.4 < k < 10.

5k – 12 > 0

k < 2.4

Substitute the given values.

Add 12 to both sides and divide by 5.

Check It Out! Example 1a

Compare m∠EGH and m∠EGF.

Compare the side lengths in ∆EGH and ∆EGF.

FG = HG EG = EG EF > EH

By the Converse of the Hinge Theorem, m∠EGH < m∠EGF.

Check It Out! Example 1b

Compare BC and AB.

Compare the side lengths in ∆ABD and ∆CBD.

By the Hinge Theorem, BC > AB.

AD = DC BD = BD m∠ADB > m∠BDC.

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:

Statements Reasons

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.

Prove: AB > ED

m∠1 = m∠2 m∠3 > m∠4

1. Given

2. Def. of Midpoint

3. Def. of ≅ ∠s

4. Conv. of Isoc. ∆ Thm.

5. Hinge Thm.

1. C is the mdpt. of BDm∠3 > m∠4, m∠1 = m∠2

3. ∠1 ≅ ∠2

5. AB > ED

Statements Reasons

Proof:

Write a two-column proof.

Given:

Prove: m∠TSU > m∠RSU

Statements Reasons

1. Given

3. Reflex. Prop. of ≅

4. Conv. of Hinge Thm.

2. Conv. of Isoc. Δ Thm.

1. ∠SRT ≅ ∠STRTU > RU

∠SRT ≅ ∠STRTU > RU

Check It Out! Example 3b

4. m∠TSU > m∠RSU

Lesson Quiz: Part I

1. Compare m∠ABC and m∠DEF.

2. Compare PS and QR.

m∠ABC > m∠DEF

PS < QR

Lesson Quiz: Part II

3. Find the range of values for z.

–3 < z < 7

Statements Reasons

1. Given

2. Reflex. Prop. of ≅

3. Conv. of Hinge Thm.3. m∠XYW < m∠ZWY

Given:

Prove: m∠XYW < m∠ZWY

4. Write a two-column proof.

Lesson Quiz: Part III

Proof:

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