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4-7
Common Core State Standards
Congruence in
Overlapping Triangles
G-SRT.B.5 Use congruence . . . criteria to solve
problems and prove relationships in geometric figures.
MP 1, MP 3, MP 4
Objectives To identify congruent overlapping triangles
To prove two triangles congruent using other congruent triangles
Do all the triangles
make you dizzy?
Try to see each
one. Then learn
some tricks that
may help you.
An assignment for your graphic design class is to
make a colorful design using triangles. How many
triangles are in your design? Explain how you
count them.
MATHEMATICAL
PRACTICES In the Solve It, you located individual triangles among a jumble of triangles. Some
triangle relationships are difficult to see because the triangles overlap.
Essential Understanding You can sometimes use the congruent corresponding
parts of one pair of congruent triangles to prove another pair of triangles congruent.
This often involves overlapping triangles.
Overlapping triangles may have a common side or angle. You can simplify your work
with overlapping triangles by separating and redrawing the triangles.
Problem 1
How can you see an
individual triangle in
order to redraw it?
Use your finger to
trace along the lines
connecting the three
vertices. Then cover up
any untraced lines.
Identifying Common Parts
What common angle do △ACD and
△ECB share?
B
Separate and redraw △ACD and △ECB.
A
E
A
D
C
E
D
B
C
C
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The common angle is ∠C.
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Got It? 1.a.What is the common side in △ABD and
B
C
△DCA?
b.What is the common side in △ABD and
△BAC?
A
D
265
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Lesson 4-7 Congruence in Overlapping Triangles
Proof
Problem 2
Using Common Parts
Z
Given: ∠ZXW ≅ ∠YWX, ∠ZWX ≅ ∠YXW
Y
Prove: ZW ≅ YX
W
X
•∠ZXW ≅ ∠YWX and ∠ZWX ≅ ∠YXW
•The diagram shows that △ZWX and △YXW are
overlapping triangles.
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Show △ZWX ≅ △YXW . Then use
corresponding parts of congruent
triangles to prove ZW ≅ YX.
A diagram of the triangles separated
Z
W
Y
XW
X
∠ZXW ≅ ∠YWX
Given
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△ZWX ≅ △YXW
WX ≅ WX
Reflexive Prop. of ≅
ZW ≅ YX
ASA
Corresp. parts of
≅ are ≅.
∠ZWX ≅ ∠YXW
Given
A
2.Given: △ACD ≅ △BDC
Got It?
Prove: CE ≅ DE
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Proof
How do you choose
another pair of
triangles to help in
your proof?
Look for triangles that
share parts with △GED
and △JEB and that you
can prove congruent.
In this case, first prove
△AED ≅ △CEB.
Problem 3
B
E
C
D
Using Two Pairs of Triangles
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Given: In the origami design, E is the midpoint of AC and DB.
Prove: △GED ≅ △JEB
Proof: E is the midpoint of AC and DB, so AE ≅ CE
and DE ≅ BE. ∠AED ≅ ∠CEB because vertical
angles are congruent. Therefore, △AED ≅ △CEB
by SAS. ∠D ≅ ∠B because corresponding
parts of congruent triangles are congruent.
∠GED ≅ ∠JEB because vertical angles are
congruent. Therefore, △GED ≅ △JEB by ASA.
A
G
D
266
Chapter 4 Congruent Triangles
B
E
J
C
3.Given: PS ≅ RS, ∠PSQ ≅ ∠RSQ
Got It?
Prove: △QPT ≅ △QRT
Q
P
R
T
S
When several triangles overlap and you need to use one pair of congruent triangles to
prove another pair congruent, you may find it helpful to
draw a diagram of each pair of
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triangles.
Proof
Problem 4
Separating Overlapping Triangles
Given: CA ≅ CE , BA ≅ DE
C
Prove: BX ≅ DX
B
X
E
A
Which triangles are
useful here?
If △BXA ≅ △DXE,
then BX ≅ DX
because they are
corresponding parts. If
△BAE ≅ △DEA,
you will have enough
information to show
△BXA ≅ △DXE.
B
X
D
A
B
E
Statements
D
D
A
E
E
A
Reasons
1) BA ≅ DE
1) Given
3) ∠CAE ≅ ∠CEA
3) Base ⦞ of an isosceles △ are ≅.
4) AE ≅ AE
4) Reflexive Property of ≅
5) △BAE ≅ △DEA
5) SAS
6) ∠ABE ≅ ∠EDA
s are ≅.
6) Corresp. parts of ≅ △
7) ∠BXA ≅ ∠DXE
7) Vertical angles are ≅.
8) △BXA ≅ △DXE
8) AAS
9) BX ≅ DX
s are ≅.
9) Corresp. parts of ≅ △
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2)
CA ≅ CE
2) Given
4.Given: ∠CAD ≅ ∠EAD, ∠C ≅ ∠E
Got It?
Prove: BD ≅ FD
B
A
C
D
F
E
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Lesson 4-7 Congruence in Overlapping Triangles
267
Lesson Check
MATHEMATICAL
Do you know HOW?
Do you UNDERSTAND?
Identify any common angles or sides.
5.Reasoning In Exercise 1, both triangles have vertices
J and K. Are ∠J and ∠K common angles for △MKJ
and △LJK ? Explain.
1.△MKJ and △LJK M
2.△DEH and △DFG
E
G
L
6.Error Analysis In the diagram,
△PSY ≅ △SPL. Based on that
fact, your friend claims that
△PRL R △SRY . Explain why
your friend is incorrect.
D
J
H
K
PRACTICES
F
Separate and redraw the overlapping triangles. Label
the vertices.
P
S
R
Y
L
7.In the figure below, which pair of triangles could
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3. N
4.
B
P
△ACD ≅ △CAB? Explain.
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E
D
B
A
A
Q
M
C
E
D
C
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MATHEMATICAL
Practice and Problem-Solving Exercises
A Practice
PRACTICES
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8.
K
P
L
9.
D
E
10. X
N
F
Z
Y
M
Separate and redraw the indicated triangles. Identify any common angles
or sides.
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11.
△PQS and △QPR
12.△ACB and △PRB
P
A
Q
P
T
S
B
O
J
M
R
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Chapter 4 Congruent Triangles
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13.△JKL
and △MLK
K
L
C
R
268
T
W
G
See Problem 1.
In each diagram, the red and blue triangles are congruent. Identify their
common side or angle.
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14.Developing Proof Complete the flow proof.
See Problem 2.
P
Given: ∠T ≅ ∠R, PQ ≅ PV
Prove: ∠PQT ≅ ∠PVR
Q
V
S
∠T ≅ ∠R
R
T
a.
∠TPQ ≅ ∠RPV
△TPQ ≅ △RPV
b.
d.
∠PQT ≅ ∠PVR
e.
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PQ ≅ PV
c.
15.Given: RS ≅ UT , RT ≅ US
Proof
Prove: △RST ≅ △UTS
T
S
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M
R
R
D
V
17.Given: ∠1 ≅ ∠2, ∠3 ≅ ∠4
Proof
Prove: △QET ≅ △QEU
T
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3
4
Prove: △QDA ≅ △UAD
U
Q
U
W
Q
16.Given: QD ≅ UA, ∠QDA ≅ ∠UAD
Proof
1
2
E
A
See Problems 3 and 4.
18.Given: AD ≅ ED,
D is the midpoint of BF
Prove: △ADC ≅ △EDG
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A
G
Proof
B
F
U
E
B
D
C
19.
Think About a Plan In the diagram at the right, ∠V ≅ ∠S,
Q
P
B Apply
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VU ≅ ST, and PS ≅ QV. Which two triangles are congruent
X
by SAS? Explain.
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W
R
• How can you use a new diagram to help you identify
the triangles?
V
S
U
T
• What do you need to prove triangles congruent by SAS?
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Lesson 4-7 Congruence in Overlapping Triangles
269
STEM
20.Clothing
Design The figure at the right is part of a clothing
design pattern, and it has the following relationships.
• GC # AC
• AB # BC
• AB } DE } FG
• m∠A = 50
• △DEC is isosceles with base DC.
G
A
a.
Find the measures of all the numbered angles in the figure.
b.
Suppose AB ≅ FC. Name two congruent triangles and
explain how you can prove them congruent.
D
2
F
3
6
C
Proof
Q
P
D
F
A
1
5
22.Given: QT # PR, QT bisects PR,
QT bisects ∠VQS
Prove: VQ ≅ SQ 21.Given: AC ≅ EC , CB ≅ CD
Proof
Prove: ∠A ≅ ∠E
C
B
7
E
B
J
4 H 8 9
I
V
E
R
S
T
Open-Ended Draw the diagram described.
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23.
Draw a vertical segment on your paper. On the right side of the segment draw two
triangles that share the vertical segment as a common side.
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24.
Draw two triangles that have a common angle.
25.Given: TE ≅ RI , TI ≅ RE,
∠TDI and ∠ROE are right ⦞
Proof
Prove: TD ≅ RO
T
26.Given: AB # BC , DC # BC,
AC ≅ DB
Proof
Prove: AE ≅ DE
I
A
O
E
D
E
C Challenge
D
R
27.Identify a pair of overlapping congruent triangles in the
diagram. Then use the given information to write a proof
to show that the triangles are congruent.
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Given: AC ≅ BC, ∠A ≅ ∠B
C
B
A
B
F
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E
D
C
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270
Chapter 4 Congruent Triangles
28.Reasoning Draw a quadrilateral ABCD with AB } DC, AD } BC, and diagonals AC
and DB intersecting at E. Label your diagram to indicate the parallel sides.
a.
List all the pairs of congruent segments in your diagram.
b.
Writing Explain how you know that the segments you listed are congruent.
Proof
Standardized Test Prep
SAT/ACT
29.According to the diagram at the right, which statement is true?
△DEH ≅ △GFH by AAS
△DEH ≅ △GFH by SAS
△DEF ≅ △GFE by AAS
△DEF ≅ △GFE by SAS
Extended
Response
F
H
E
30.△ABC is isosceles with base AC. If m∠C = 37, what is m∠B?
37
G
74
D
106
143
31.Which word correctly completes the statement “All ? angles are congruent”?
adjacent
supplementary
right
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corresponding
J
32.In the figure, LJ } GK and M is the midpoint of LG.
a.
Copy the diagram. Then mark your diagram with the given information.
b.
Prove △LJM ≅ △GKM.
c.
Can you prove that △LJM ≅ △GKM another way? Explain.
G
M
L
K
Mixed Review
See Lesson 4-6.
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33.
Developing Proof Complete the paragraph proof.
Given: AB ≅ DB, ∠A and ∠D are right angles
D
Prove: △ABC ≅ △DBC
C
Proof: You are given that AB ≅ DB and ∠A and ∠D are right
angles. △ABC and △DBC are a. ? triangles by the definition of
b. ? triangle. BC ≅ BC by the c. ? Property of Congruence.
△ABC ≅ △DBC by the d. ? Theorem.
B
34.
Constructions Draw a line p and a point M not on p. Then construct line n
through M so that n # p.
Get Ready! To prepare for Lesson 5-1, do Exercises 35–37.
Find the coordinates of the midpoint of AB.
35.
A( -2, 3), B(4, 1)
36.A(0, 5), B(3, 6)
A
See Lesson 3-6.
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See Lesson 1-7.
37.A(7, 10), B( -5, -8)
Lesson 4-7 Congruence in Overlapping Triangles
271