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5-6
5-6 Inequalities
InequalitiesininTwo
TwoTriangles
Triangles
Warm Up
Lesson Presentation
Lesson Quiz
HoltMcDougal
GeometryGeometry
Holt
5-6 Inequalities in Two Triangles
Warm up: Determine if it is possible to construct
a triangle with the following vertices
1. (1,1)(2,2)(3,3)
2.
No- straight
line
Yes
3.
x=4
A=65 B=73
So… AB, BC, AC
Holt McDougal Geometry
C=42
5-6 Inequalities in Two Triangles
Warm Up
4. Write the angles in order from smallest to
largest.
X, Z, Y
5. The lengths of two sides of a triangle are 12 cm
and 9 cm. Find the range of possible lengths
for the third side. 3 cm < s < 21 cm
Holt McDougal Geometry
5-6 Inequalities in Two Triangles
Objective
Apply inequalities in two triangles.
Holt McDougal Geometry
5-6 Inequalities in Two Triangles
Holt McDougal Geometry
5-6 Inequalities in Two Triangles
Example 1: Using the Hinge Theorem and Its
Converse
Compare mBAC and mDAC.
By the Converse of the Hinge Theorem,
mBAC > mDAC.
Holt McDougal Geometry
5-6 Inequalities in Two Triangles
Example 2: Using the Hinge Theorem and Its
Converse
Compare EF and FG.
Compare the sides and angles in
∆EFH angles in ∆GFH.
mGHF = 180° – 82° = 98°
By the Hinge Theorem, EF < GF.
Holt McDougal Geometry
98
5-6 Inequalities in Two Triangles
Example 3: Using the Hinge Theorem and Its
Converse
Find the range of values for k.
Step 1 Compare the side
lengths in ∆MLN and ∆PLN.
LN = LN
LM = LP
MN > PN
By the Converse of the Hinge Theorem,
mPLN < mMLN
5k – 12 < 38
k < 10
Holt McDougal Geometry
Substitute the given values.
Add 12 to both sides and divide by 5.
5-6 Inequalities in Two Triangles
What else do we know about the mPLN
Step 2 Since PLN is in a triangle, mPLN > 0°.
5k – 12 > 0
k < 2.4
Step 3 Combine the two inequalities.
The range of values for k is 2.4 < k < 10.
Holt McDougal Geometry
5-6 Inequalities in Two Triangles
Check It Out! Example 4
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.
Holt McDougal Geometry
5-6 Inequalities in Two Triangles
Check It Out! Example 5
Compare BC and AB.
Compare the side lengths in ∆ABD
and ∆CBD.
AD = DC
BD = BD
mADB > mBDC.
By the Hinge Theorem, BC > AB.
Holt McDougal Geometry
5-6 Inequalities in Two Triangles
Example 6: 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 SE. Who is farther
from school? Explain.
Holt McDougal 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 McDougal Geometry
5-6 Inequalities in Two Triangles
PROOF #1
Write a two-column proof.
Given: C is the midpoint of BD.
m1 = m 2
m3 > m 4
Prove: AB > ED
Holt McDougal Geometry
5.6 Indirect Proof and Inequalities in Two Triangles
5-6 Inequalities in Two Triangles
Proof:
Statements
1. C is the mdpt. of BD
m3 > m4,
m1 = m2
Reasons
1. Given
2. Def. of Midpoint
3. 1  2
3. Def. of  s
4. Conv. of Isoc. ∆ Thm.
5. AB > ED
Holt McDougal Geometry
5. Hinge Thm.
5-6 Inequalities in Two Triangles
Proof #2: 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 McDougal Geometry
5-6 Inequalities in Two Triangles
Lesson Quiz
1. Compare mABC and mDEF.
mABC > mDEF
2. Compare PS and QR.
PS < QR
Holt McDougal Geometry
5-6 Inequalities in Two Triangles
Lesson Quiz
3. Find the range of values for z.
–3 < z < 7
Holt McDougal 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 McDougal Geometry
5-6 Inequalities in Two Triangles
Holt McDougal Geometry
5-6 Inequalities in Two Triangles
Indirect Proof #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 McDougal Geometry
5-6 Inequalities in Two Triangles
Check It Out! Example 1 Continued
Step 3 Use direct reasoning to lead to a contradiction.
m1 + m2 + m3 = 180°
90° + 90° + m3 = 180°
180° + m3 = 180°
m3 = 0°
However, by the Protractor Postulate, a triangle
cannot have an angle with a measure of 0°.
Holt McDougal Geometry
5-6 Inequalities in Two Triangles
Check It Out! Example 1 Continued
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.
Holt McDougal Geometry