5-6

5-6 Inequalities in Two Triangles

Holt Geometry

Warm UpWarm UpLesson PresentationLesson PresentationLesson QuizLesson Quiz

Warm Up1. 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

5.6 Indirect Proof and Inequalities in Two Triangles

Apply inequalities in two triangles.Objective


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


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°


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 < 38k < 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, mPLN > 0°.

Step 3 Combine the two inequalities.

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

5k – 12 > 0k < 2.4

Substitute the given values.Add 12 to both sides and divide by 5.


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.


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.


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 Reasons1. 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

m1 = m2 m3 > m4


1. Given

2. Def. of Midpoint3. Def. of s4. Conv. of Isoc. ∆ Thm.5. Hinge Thm.

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

3. 1 2

5. AB > ED

Statements ReasonsProof:


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


Lesson Quiz: Part I

1. Compare mABC and 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


Statements Reasons1. Given

2. Reflex. Prop. of 3. Conv. of Hinge Thm.3. mXYW < mZWY

Given:Prove: mXYW < mZWY4. Write a two-column proof.

Lesson Quiz: Part III

Proof:


5-6

Documents

Transcript of 5-6