Holt Geometry

5-6 Inequalities in Two Triangles5-6 Inequalities in Two Triangles

Holt Geometry

Warm Up

Lesson Presentation

Lesson Quiz

Holt Geometry

5-6 Inequalities in 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

Holt Geometry

5-6 Inequalities in Two Triangles

Apply inequalities in two triangles.

Write an indirect proof.

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

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

Holt Geometry

5-6 Inequalities in Two Triangles

So far you have written proofs using direct reasoning. You began with a true hypothesis and built a logical argument to show that a conclusion was true.

In an indirect proof, you begin by assuming that the conclusion is false. Then you show that this assumption leads to a contradiction. This type of proof is also called a proof by contradiction.

Holt Geometry

5-6 Inequalities in Two Triangles

Holt Geometry

5-6 Inequalities in Two Triangles

When writing an indirect proof, look for a contradiction of one of the following: the given information, a definition, a postulate, or atheorem.

Helpful Hint

Holt Geometry

5-6 Inequalities in Two Triangles

Example 1: Writing an Indirect Proof

Step 1 Identify the conjecture to be proven.

Given: a > 0

Step 2 Assume the opposite of the conclusion.

Write an indirect proof that if a > 0, then

Prove:

Assume

Holt Geometry

5-6 Inequalities in Two Triangles

Example 1 Continued

Step 3 Use direct reasoning to lead to a contradiction.

However, 1 > 0.

1 0

Given, opposite of conclusion

Zero Prop. of Mult. Prop. of Inequality

Simplify.

Holt Geometry

5-6 Inequalities in Two Triangles

Step 4 Conclude that the original conjecture is true.

Example 1 Continued

The assumption that is false.

Therefore

Holt Geometry

5-6 Inequalities in Two Triangles

Check It Out! Example 1

Write an indirect proof that a triangle cannot have two right angles.

Step 1 Identify the conjecture to be proven.

Given: A triangle’s interior angles add up to 180°.

Prove: A triangle cannot have two right angles.

Step 2 Assume the opposite of the conclusion.

An angle has two right angles.

Holt Geometry

5-6 Inequalities in Two Triangles

Check It Out! Example 1 Continued

Step 3 Use direct reasoning to lead to a contradiction.

However, by the Protractor Postulate, a triangle cannot have an angle with a measure of 0°.

m1 + m2 + m3 = 180°

90° + 90° + m3 = 180°

180° + m3 = 180°

m3 = 0°

Holt Geometry

5-6 Inequalities in Two Triangles

Step 4 Conclude that the original conjecture is true.

The assumption that a triangle can have two right angles is false.

Therefore a triangle cannot have two right angles.

Check It Out! Example 1 Continued

Holt Geometry

5-6 Inequalities in Two Triangles

Lesson Quiz: Part I

1. Compare mABC and mDEF.

2. Compare PS and QR.

mABC > mDEF

PS < QR

Holt Geometry

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

Statements Reasons

1. Given

2. Reflex. Prop. of

3. Conv. of Hinge Thm.3. mXYW < mZWY

Given:

Prove: mXYW < mZWY

4. Write a two-column proof.

Lesson Quiz: Part III

Proof: