Holt Geometry

5-6 Inequalities in Two Triangles

Apply inequalities in two triangles.

Objective

Holt Geometry

5-6 Inequalities in Two Triangles

Holt Geometry

5-6 Inequalities in Two Triangles

Example 1A: Using the Hinge Theorem and Its Converse

Compare mBAC and mDAC.

Compare the side lengths in ∆ABC and ∆ADC.

By the Converse of the Hinge Theorem, mBAC > mDAC.

AB = AD AC = AC BC > DC

Holt Geometry

5-6 Inequalities in Two Triangles

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 mEHF > mGHF

mGHF = 180° – 82° = 98°

Holt Geometry

5-6 Inequalities in Two Triangles

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, mMLN > mPLN.

LN = LN LM = LP MN > PN

5k – 12 < 38

k < 10

Substitute the given values.

Add 12 to both sides and divide by 5.

Holt Geometry

5-6 Inequalities in Two Triangles

Example 1C Continued

Step 2 Since PLN is in a triangle, mPLN > 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.

Holt Geometry

5-6 Inequalities in Two Triangles

Check It Out! Example 1a

Compare mEGH and mEGF.

Compare the side lengths in ∆EGH and ∆EGF.

FG = HG EG = EG EF > EH

By the Converse of the Hinge Theorem, mEGH < mEGF.

Holt Geometry

5-6 Inequalities in Two Triangles

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 mADB > mBDC.

Holt Geometry

5-6 Inequalities in Two Triangles

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.

Holt Geometry

5-6 Inequalities in Two Triangles

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.

Holt Geometry

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 proof.

Given:

Prove: AB > 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 3a

Write a two-column proof.

Given: C is the midpoint of BD.

Prove: AB > ED

m1 = m2

m3 > m4

Holt Geometry

5-6 Inequalities in Two Triangles

1. Given

2. Def. of Midpoint

3. Def. of s

4. Conv. of Isoc. ∆ Thm.

5. Hinge Thm.

1. C is the mdpt. of BDm3 > m4, m1 = m2

3. 1 2

5. AB > ED

Statements Reasons

Proof:

Holt Geometry

5-6 Inequalities in Two Triangles

Write a two-column proof.

Given:

Prove: mTSU > mRSU

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. mTSU > mRSU