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Transcript of 5-6
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
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
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