Inequalities in Two Triangles

1
5-6
Inequalities in Two Triangles
Warm Up
Lesson Presentation
Lesson Quiz
Holt Geometry
2
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 lt s lt 21 cm
3
Objective
Apply inequalities in two triangles.
4
(No Transcript)
5
Example 1A Using the Hinge Theorem and Its
Converse
Compare m?BAC and m?DAC.
Compare the side lengths in ?ABC and ?ADC.
AB AD AC AC BC gt DC
By the Converse of the Hinge Theorem, m?BAC gt
m?DAC.
6
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 82 98
EH GH FH FH m?EHF gt m?GHF
By the Hinge Theorem, EF lt GF.
7
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
Substitute the given values.
k lt 10
Add 12 to both sides and divide by 5.
8
Example 1C Continued
Step 2 Since ?PLN is in a triangle, m?PLN gt 0.
5k 12 gt 0
Substitute the given values.
k lt 2.4
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.
9
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.
10
Check It Out! Example 1b
Compare BC and AB.
Compare the side lengths in ?ABD and
?CBD.
AD DC BD BD m?ADB gt m?BDC.
By the Hinge Theorem, BC gt AB.
11
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.
12
Example 2 Continued
The distances of 3 blocks and 4 blocks are the
same in both triangles.
The angle formed by Johns route (90º) is smaller
than the angle formed by Lukes route (100º). So
Luke is farther from school than John by the
Hinge Theorem.
13
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.
14
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.
15
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
16
Proof
Statements Reasons

1. Given
1. C is the mdpt. of BD m?3 gt m?4, m?1 m?2
2. Def. of Midpoint
3. Def. of ? ?s
3. ?1 ? ?2
4. Conv. of Isoc. ? Thm.
5. Hinge Thm.
5. AB gt ED
17
Check It Out! Example 3b
Write a two-column proof.
Given
?SRT ? ?STR TU gt RU
Prove m?TSU gt m?RSU
Statements Reasons

1. Given
1. ?SRT ? ?STR TU gt RU
2. Conv. of Isoc. ? Thm.
3. Reflex. Prop. of ?
4. m?TSU gt m?RSU
4. Conv. of Hinge Thm.
18
Lesson Quiz Part I
1. Compare m?ABC and m?DEF. 2. Compare PS
and QR.
m?ABC gt m?DEF
PS lt QR
19
Lesson Quiz Part II
3. Find the range of values for z.
3 lt z lt 7
20
Lesson Quiz Part III
4. Write a two-column proof.
Prove m?XYW lt m?ZWY
Given
Proof
Statements Reasons