# GEOMETRY 5-6 ```5-6
5-6 Inequalities
InequalitiesininTwo
TwoTriangles
Triangles
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Geometry
Holt
Geometry
5-6 Inequalities in Two Triangles
Warm Up
1. Write the angles in order from smallest to
largest.
∠X, ∠Z, ∠Y
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. 3 cm &lt; s &lt; 21 cm
Holt Geometry
5-6 Inequalities in Two Triangles
Objective
Apply inequalities in two triangles.
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 m∠BAC and m∠DAC.
Compare the side lengths in ∆ABC
AC = AC
BC &gt; DC
By the Converse of the Hinge Theorem,
m∠BAC &gt; m∠DAC.
Holt Geometry
5-6 Inequalities in Two Triangles
Example 1B: Using the Hinge Theorem and Its
Converse
Compare EF and FG.
Compare the sides and angles in
∆EFH angles in ∆GFH.
m∠GHF = 180&deg; – 82&deg; = 98&deg;
EH = GH
FH = FH
m∠EHF &gt; m∠GHF
By the Hinge Theorem, EF &lt; GF.
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.
LN = LN
LM = LP
MN &gt; PN
By the Converse of the Hinge Theorem,
m∠MLN &gt; m∠PLN.
5k – 12 &lt; 38
k &lt; 10
Holt Geometry
Substitute the given values.
Add 12 to both sides and divide by 5.
5-6 Inequalities in Two Triangles
Example 1C Continued
Step 2 Since ∠PLN is in a triangle, m∠PLN &gt; 0&deg;.
5k – 12 &gt; 0
k &lt; 2.4
Substitute the given values.
Add 12 to both sides and divide by 5.
Step 3 Combine the two inequalities.
The range of values for k is 2.4 &lt; k &lt; 10.
Holt Geometry
5-6 Inequalities in Two Triangles
Check It Out! Example 1a
Compare m∠EGH and m∠EGF.
Compare the side lengths in ∆EGH
and ∆EGF.
FG = HG
EG = EG
EF &gt; EH
By the Converse of the Hinge Theorem,
m∠EGH &lt; m∠EGF.
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.
BD = BD
By the Hinge Theorem, BC &gt; AB.
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 10&ordm; 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 (90&ordm;) is smaller than the
angle formed by Luke’s route
(100&ordm;). 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 &gt; 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.
m∠1 = m∠2
m∠3 &gt; m∠4
Prove: AB &gt; ED
Holt Geometry
5-6 Inequalities in Two Triangles
Proof:
Statements
1. C is the mdpt. of BD
m∠3 &gt; 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 &gt; ED
Holt Geometry
5. Hinge Thm.
5-6 Inequalities in Two Triangles
Check It Out! Example 3b
Write a two-column proof.
Given: ∠SRT ≅ ∠STR
TU &gt; RU
Prove: m∠TSU &gt; m∠RSU
Statements
1. ∠SRT ≅ ∠STR
TU &gt; RU
Reasons
1. Given
2. Conv. of Isoc. Δ Thm.
3. Reflex. Prop. of ≅
4. m∠TSU &gt; m∠RSU
Holt Geometry
4. Conv. of Hinge Thm.
5-6 Inequalities in Two Triangles
Lesson Quiz: Part I
1. Compare m∠ABC and m∠DEF.
m∠ABC &gt; m∠DEF
2. Compare PS and QR.
PS &lt; QR
Holt Geometry
5-6 Inequalities in Two Triangles
Lesson Quiz: Part II
3. Find the range of values for z.
–3 &lt; z &lt; 7
Holt Geometry
5-6 Inequalities in Two Triangles
Lesson Quiz: Part III
4. Write a two-column proof.
Prove: m∠XYW &lt; m∠ZWY
Given:
Proof:
Statements
Reasons
1. Given
2. Reflex. Prop. of ≅
3. m∠XYW &lt; m∠ZWY
Holt Geometry
3. Conv. of Hinge Thm.
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