Page 1 of 7
5.5
Goal
Show corresponding parts
of congruent triangles are
congruent.
Key Words
• corresponding parts
p. 233
Using Congruent Triangles
If you know that two triangles are congruent, you can use the
definition of congruent triangles from Lesson 5.1 to conclude
that the corresponding parts are congruent.
If you know TABC c TDEF, then you can conclude:
Corresponding Sides
Corresponding Angles
&* c DE
&**
AB
aA c aD
&* c EF
&*
BC
aB c aE
&* c DF
&**
AC
aC c aF
C
A
B
D
E
F
EXAMPLE
1
Use Corresponding Parts
&* and CD
&* bisect
In the diagram, AB
each other at M. Prove that aA c aB.
C
A
M
B
D
Solution
1 First sketch the diagram and label any
●
congruent segments and congruent
angles.
C
A
2 Because aA and aB are corresponding
●
M
angles in TADM and TBCM, show
that TADM c TBCM to prove that
aA c aB.
D
Statements
Reasons
&* and CD
&* bisect
1. AB
1. Given
B
each other at M.
&** c MB
&**
2. MA
2. Definition of segment bisector
3. aAMD c aBMC
3. Vertical Angles Theorem
Student Help
&** c MC
&**
4. MD
4. Definition of segment bisector
LOOK BACK
5. TADM c TBCM
5. SAS Congruence Postulate
6. aA c aB
6. Corresponding parts of congruent
For the definition of
congruent figures,
see p. 233.
triangles are congruent.
5.5
Using Congruent Triangles
265
Page 2 of 7
Diagrams often show overlapping triangles. It is usually helpful to
visualize or redraw the triangles so that they do not overlap.
Original diagram
Redrawn diagram
B
D
B
C
A
C D
E
A
TABC and TEBD overlap.
EXAMPLE
2
Visualize Overlapping Triangles
Student Help
Drawing overlapping
triangles separately
makes it easier to see
the triangles and
correctly mark
congruent sides and
angles, as shown on
p. 232.
E
TABC and TEBD do not overlap.
Sketch the overlapping triangles separately. Mark all
congruent angles and sides. Then tell what theorem
or postulate you can use to show TJGH c TKHG.
VISUAL STRATEGY
B
J
K
G
H
Solution
1 Sketch the triangles separately and mark any given information.
●
Think of TJGH moving to the left and TKHG moving to the right.
K
J
Mark aGJH c aHKG and
aJHG c aKGH.
G
H
G
H
2 Look at the original diagram for shared sides, shared angles,
●
or any other information you can conclude.
&** and HG
&** are the same side,
In the original diagram, GH
&**
&**
so GH c HG .
J
K
Add congruence marks to
GH
&* in each triangle.
G
H
G
H
3 You can use the AAS Congruence Theorem to show that
●
TJGH c TKHG.
266
Chapter 5
Congruent Triangles
Page 3 of 7
IStudent Help
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MORE EXAMPLES
More examples at
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EXAMPLE
3
Use Overlapping Triangles
&* c DE
&*.
Write a proof that shows AB
©
Given aABC c aDEC
&* c CE
&*
CB
Prove
A
D
E
B
© AB
&* c DE
&*
C
Solution
1 Sketch the triangles separately.
●
D
A
Then label the given information
and any other information you
can conclude from the diagram.
B
In the original diagram, aC is
the same in both triangles
(aBCA c aECD).
E
C
C
Mark aC c aC.
2 Show TABC c TDEC to prove
●
&* c DE
&*.
that AB
Statements
Reasons
1. aABC c aDEC
1. Given
&* c CE
&*
2. CB
2. Given
3. aC c aC
3. Reflexive Prop. of Congruence
4. TABC c TDEC
4. ASA Congruence Postulate
&* c DE
&*
5. AB
5. Corresponding parts of congruent
triangles are congruent.
Use Overlapping Triangles
1. Tell which triangle congruence
A
B
D
C
theorem or postulate you would
&* c CD
&*.
use to show that AB
Redraw the triangles separately and label all congruences. Explain
how to show that the triangles or corresponding parts are congruent.
& c KL
&* and
2. Given KJ
&* c ML
&**.
aJ c aL, show NJ
3. Given aSPR c aQRP and
aQ c aS, show TPQR c TRSP.
P
K
M
J
S
N
P
L
5.5
R
Using Congruent Triangles
267
Page 4 of 7
5.5 Exercises
Guided Practice
Vocabulary Check
1. In the diagram, T ABC c T DEF.
Why can you conclude that
aCBA c aFED?
Skill Check
2.
C
F
A
B E
D
Visualize It! Tell which diagram correctly represents all the
congruences in the original figure.
Original figure
D
E
G
F
A.
B.
D
E
D
E
F
E
G
F
E
F
G
F
Explain how to show that the statement is true.
3. aSTU c aUVS
S
&* c DC
&**
4. AB
T
5. aJ c aM
A
J
D
N
L
V
U
B
M
C
K
Practice and Applications
6. Showing Congruence In the
Extra Practice
P
R
diagram, T JKL c T PQR. Why
can you conclude that aK c aQ?
See p. 684.
J
P
K
L
Finding Congruent Parts Tell which triangles you need to show are
congruent in order to show that the statement is true.
Homework Help
7. aA c aD
C
Example 1: Exs. 15, 16,
19
Example 2: Exs. 10–13
Example 3: Exs. 14–20
Chapter 5
&* c BA
&*
9. DE
K
J
C
F
L
A
268
8. aJ c aN
Congruent Triangles
B
D
M
N
A
D
B
E
Page 5 of 7
Visualize It! Sketch the overlapping triangles separately. Mark all
congruent angles and sides. Then tell what theorem or postulate you
can use to show that the triangles are congruent.
&* c DA
&*, aADB c aCBD
10. BC
&* c HJ
&*
11. aE c aH, EF
D
B
E
J
G
C
A
Crafts
F
H
String Designs What theorem or postulate can you use to show
that the triangles in the string design are congruent? Explain your
reasoning.
12. T ABC c TDEF
13. TGHJ c T KHM
C
H
F
J
A
E
B
D
M
K
G
STRING DESIGNS The
shape and size of a string
design is determined by how
many points along a circle are
used to create the design.
14. Logical Reasoning Fill in the missing statements and reasons.
Given
© AB
&* c AE
&*
aACB c aADE
E
C
F
Prove © aB c aE
D
A
B
Statements
Reasons
&* c AE
&*
1. AB
1. _________?_________
2. _________?_________
2. Given
3. _________?_________
3. Reflexive Prop. of Congruence
4. T ABC c T AED
4. _________?_________
5. aB c aE
5. _________?_________
Finding Congruent Parts Use the information in the diagram to
prove that the statement is true.
15. aA c aC
A
B
&* c NM
&&
16. JK
C
&* c SR
&*
17. QT
R
J
T
M
L
K
P
D
U
N
5.5
Using Congruent Triangles
S
269
Page 6 of 7
18. Argyle Patterns In the argyle pattern shown below,
IStudent Help
&** c VW
&** c XY
&* c YZ
&* and aUVW c aXYZ. Prove that
UV
&& c ZX
&*.
WU
ICLASSZONE.COM
HOMEWORK HELP
Extra help with problem
solving in Ex. 18 is
at classzone.com
W
Y
U
Z
V
X
Logical Reasoning In Exercises 19 and 20, fill in the missing
statements and reasons.
19.
&* and AE
&* bisect each other at C.
Given © BD
Prove © aA c aE
E
C
B
A
Statements
Reasons
&* and AE
&* bisect each
1. BD
1. _________?_________
other at C.
Student Help
20.
&* c DC
&*
2. BC
2. _________?_________
3. _________?_________
3. Def. of segment bisector
4. aBCA c aDCE
4. _________?_________
5. TABC c TEDC
5. _________?_________
6. aA c aE
6. _________?_________
Given
© JK
&* ∏ LK
&*, ML
&
& ∏ KL
&*
K
L
J
M
& c MK
&&
JL
STUDY TIP
If you get stuck,
remember that you
know the given
information and what
you are trying to prove.
You can fill those in
first, then go back to
the other steps.
&* c ML
&
&
Prove © JK
Statements
Reasons
1. _________?_________
1. Given
&* ∏ LK
&*, ML
&
& ∏ KL
&*
2. JK
2. _________?_________
3. _________?_________
3. ∏ lines form right angles.
4. TJKL and TMLK are
4. _________?_________
right triangles.
270
Chapter 5
5. _________?_________
5. Reflexive Prop. of Congruence
6. TJKL c TMLK
6. _________?_________
7. _________?_________
7. _________?_________
Congruent Triangles
D
Page 7 of 7
&* i PN
&*,
21. Challenge In the figure at the right, LK
&* c ML
&* c LP
&*. What
aLJK c aPMN, and JM
theorem or postulate can be used to show
that TJKL c TMNP? Explain.
Standardized Test
Practice
N
K
J
M
L
P
&* c CB
&* and
22. Multiple Choice In the diagram, suppose that AD
aBCA c aDAC. Which triangles can you use to prove that
aEBA c aEDC ?
A
X
TABC and TCDA
B
X
C
X
TABE and TCDE
D
X
Not enough information
A
B
E
TDEB and TAEC
D
C
&* c CE
&* and
23. Multiple Choice In the diagram, suppose that AE
&* c DE
&*. Which triangles can you use to prove that AB
&* c CD
&** ?
BE
F
X
G
X
H
X
J
X
Mixed Review
TABC and TCDA
A
B
TABE and TCDE
E
TDEB and TAEC
D
Not enough information
C
&( bisects aABC. Find the
Angle Bisectors In the diagram, BD
value of x. (Lesson 2.2)
24.
25.
A
A
3x 8
D
(x 2 3)8
A
D
258
B
26.
548
C
C
B
B
(5x 1 5)8
D
(6x 2 1)8
C
Perpendicular Lines Find the value of x, given that p ∏ q. (Lesson 3.2)
p
27.
28. p
29.
p
488
x8
Algebra Skills
q
(8x 1 2)8
378
(4x 1 1)8
q
q
Solving Equations Solve the equation. (Skills Review, p. 673)
30. x 1 5 5 8
31. 7x 5 263
32. 4x 2 9 5 23
33. 11 1 3x 5 32
34. 5x 2 3x 1 10 5 24
35. x 1 2x 2 8 5 19
5.5
Using Congruent Triangles
271