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Using Corresponding Parts of
Congruent Triangles
What You'll Learn
@ Check Skills You'll Need
• To identify congruent
overlapping triangles
for Help '~
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Some triangle relationships are difficult to see because the triangles overlap.
Overlapping triangles may have a common side or angle. You can simplify your
work with overlapping triangles by separating and redrawing the triangles.
Overlapping triangles
share part or all of one or
more sides.
A
Identifying
Separate and
redraw L.DFG and
L.EHG. Identify
the common angle.
I
•
2. Can you conclude that the triangles are congruent? Explain.
a. L.AZK and L.DRS
b. L.SDR and L.JTN
c. L.ZKA and L.NJT
To identify overlapping
triangles in scaffolding,
as in Example 1
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1 1 and 4-3
the next two figures in this
pattern have?
... And Why
:1><:1:
I,
I
1. How many triangles will
• To prove two triangles
congruent by first proving
two other triangles
congruent
@ Quick Check
I,
o
Common Parts
G
D
Common
angle
E
D
H~E
The diagram at the left
shows triangles from the scaffolding
that workers used when they repaired
and cleaned the Statue of Liberty.
a. Name the common side in MDC
and L.BCD.
b. Name another pair of triangles that
share a common side. Name the
common side.
Engineering
In overlapping triangles, a common side or angle is congruent to itself by the
Reflexive Property of Congruence.
Lesson 4-7
Using Corresponding
Parts of Congruent
Triangles
241
Proof
___ ..~ _ Using Common Parts
"~
Given:
LZXW
==
L YWX, LZWX
==
Z
L YXW
Write a plan and then a proof to show
that the -two "outside"
segments are congruent.
Prove: ZW == Y X
First, separate the
-overlapping
- triangles.
ZW == YX by CPCTCif
L,ZXW == L,ywx. Show
this congruence by ASA.
Plan:
y
w~x
Z
y
w~x
w~x
Proof:
I LZXW==LYWX
Given
I~
I WX == WX
•
ReflexiveProp. of ==
1
~
I
LZWX==Lyxwl/
-I
L,-Z-XW-==-L,-Y1-VX-1
ASA
/
•
Postulate
I
ZW == YX
1
CPCTC
Given
@ Quick Check
8
Write a plan and then a proof.
Given:
Prove:
A
B
c
D
-L,ACD
-DE L,BDC
CE
==
==
Sometimes you can prove one pair of triangles congruent and then use their
congruent corresponding parts to prove another pair congruent.
Proof
Using Two Pairs of Triangles
~
Given:
Prove:
In the quilt, E is the midpoint of AC and DB.
L,GED == L,JEB
Write a plan and then a proof.
Plan: L,GED == L,JEB by ASA if LD == LB. These angles are congruent by
CPCTC if MED == L,CEB. These triangles are congruent by SAS.
E is the midpoint of AC and DB, so AE == CE and DE == BE.
LAED == LCEB because vertical angles are congruent. Therefore,
MED == L,CEB by SAS. LD == LB by CPCTC, and LGED == LJEB
because they are vertical angles. Therefore, L,GED == L,JEB by ASA.
Proof:
II
@ Quick Check e Write a plan and then a proof.
Given:
PS
Prove:
L,QPT
==
RS, LPSQ
==
==
Q
p
R
LRSQ
L,QRT
s
242
Chapter 4
Congruent Triangles
••
When triangles overlap, you can keep track of information by drawing other
diagrams that separate the overlapping triangles.
Separating Overlapping Triangles
Given:
CA::= CE, BA
::=
c
DE
Write a plan and then a proof to show that two small
segments inside -the triangle are congruent.
Prove: BX::= DX
-
Plan: BX ::= DX by CPCTC if f:oBXA ::= f:oDXE.
This congruence holds by AAS if LABX ::= LEDX.
These are congruent by CPCTC in f:oBAE
and f:oDEA, which are congruent by SAS.
A
E
B
Proof:
A~E
Statements
\t' Connection
--~ Japanese paper-folding
J""" ::If origami involves many
:dapping triangles.
@QuickCheck
4)
A~E
Reasons
1. BA
::=
DE
1. Given
2. CA
::=
CE
2. Given
3. LCAE::= LCEA
3. Isosceles Triangle Theorem
4. AE
4. Reflexive Property of Congruence
5. SAS
::=
AE
5. f:oBAE ::= f:oDEA
.:.aI-World
D
6. LABE
::=
LEDA
6. CPCTC
7. LBXA
::=
LDXE
8. f:oBXA
::=
f:oDXE
7. Vertical angles are congruent.
8. AAS
9. BX
::=
DX
9.CPCTC
0 Plan a proof. Separate the overlapping
c
triangles in your plan. Then follow your
plan and write a proof.
LCAD::= LEAD,LC::=
Prove: BD::= FD
Given:
LE
A
E
EXERCISES
For more exercises, see Extra Skill, Word Problem, and Proof Practice.
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Practice by Example
Example 1
In each diagram, the red and blue triangles are congruent. Identify their common
side or angle.
(page 241)
2.
~for Help
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lesson 4-7
Using Corresponding
E
3.
ybz
x~'[
Parts of Congruent
Triangles
243
Separate
and redraw the indicated
triangles. Identify any common angles or sides.
6. 6.JKL and 6.MLK
5. 6.ACB and 6.PRB
4. 6.PQS and 6.QPR
P
s
Example 2
B
R
7. Developing Proof Complete
(page 242)
J6M
AvzP
[g]Q
P
the flow proof.
Given: L T ~ LR, PQ ~ PV
Prove: LPQT
~ LPVR
TV-
I
ILT~LRI~
a.~
LTPQ~ LRPV
b.~
PQ~W
~
---
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1/
~R
/
6.TPQ~ 6.RPVI
).
I
LPQT~ LPVR
I
e.l
d.~
c.~
-Proof
Write a plan and then a proof.
8. Given: RS ~ UT, RT ~ US
Prove: 6.RST ~ 6. UTS
S
9. Given: QD ~ UA,LQDA~LUAD
Prove: 6.QDA ~ 6.UAD
Q
T
ROU
w
Examples 3, 4
(pages 242 and 243)
10. Given:
Ll ~
U
~
D
A
V
L2, L3 ~ L4
Prove: 6.QET ~ 6.QEU
T
11. Given: AD ~ ED,
D is the midpoint of BF.
Prove: 6.ADC ~ 6.EDG
A
B
Q
U
o
E
Apply Your Skills
Open-Ended Draw the diagram described.
12. Draw a vertical segment on your paper. On the right side of the segment draw
two triangles that share the given segment as a common side.
13. Draw two triangles that have a common angle.
~.nline
Homework Video Tutor
Visit: PHSchool.com
Web Code: aue-0407
244
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Chapter 4
14. Draw two regular pentagons, each with its five diagonals.
a. In one, shade two triangles that share a common angle.
b. In the other, shade two triangles that share a common side.
15. Draw two regular hexagons and their diagonals. For these diagrams, do parts
(a) and (b) of the preceding exercise.
Congruent Triangles