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Congruent Triangles
Chapter 5
Objectives
• Identify corresponding parts of congruent triangles.
• Show triangles are congruent using the SSS, SAS,
and ASA Congruence Postulates, and the AAS and
HL Congruence Theorems.
• Use angle bisectors and perpendicular bisectors to
compute angle measures and segment lengths in
situations involving triangles.
• Reflect figures over lines and use reflections to
discover lines of symmetry in a figure.
Essential Questions
• How does the important property of
congruence relate to triangles?
• How can you identify the corresponding
parts of congruent triangles?
Sections
• 5.1 Congruence and Triangles
• 5.2 Proving Triangles are Congruent: SSS and
SAS
• 5.3 Proving Triangles are Congruent: ASA and
AAS
• 5.4 Hypotenuse-Leg Congruence Theorem: HL
• 5.5 Using Congruent Triangles
• 5.6 Angle Bisectors and Perpendicular Bisectors
• 5.7 Reflections and Symmetry
Congruence and Triangles
Section 5.1
Objectives:
• Identify congruent triangles and
corresponding parts.
Key Vocabulary
•
•
•
•
Congruent
Congruent Figures
Corresponding Parts
CPCTC
Congruence
If two geometric figures or polygons have exactly the
same shape and size, they are congruent.
• Congruent Figures
• Not Congruent
• While positioned differently,
figures 1, 2, and 3 are
exactly the same shape and
size.
• Figures 4 and 5 are exactly
the same shape, but not the
same size.
• Figures 5 and 6 are the
same size, but not exactly
the same shape.
Two figures are congruent if they are
the same size and same shape.






Congruent figures can be rotations of
one another.






Congruent figures can be reflections of
one another.





Corresponding Parts
• If two polygons are congruent, then all parts
of one polygon are congruent to the
corresponding parts (or matching parts) of
the other polygon.
• Corresponding parts include corresponding
angles and corresponding sides.
Corresponding Parts
• To name a polygon, write the vertices in
consecutive order. For example, you
can name polygon PQRS as QRSP or
SRQP, but not as PRQS.
• In a congruence statement, the order of
the vertices indicates the corresponding
parts.
Example: Congruent polygons and
Corresponding Parts
Definition of Congruent Polygons
• Two polygons are congruent if and only
if their corresponding parts are
congruent.
Then:
• Example
Given:
Congruent polygons and
Corresponding Parts
• When you write a congruence statement such as
ABC  DEF, you are also stating which parts are
congruent.
• Therefore, valid congruence statements for
congruent polygons list corresponding vertices in the
same order.
• Given the valid congruence statement ∆ABC≅∆DEF
• Other valid congruence statements; ∆BCA≅∆EFD or
∆CBA≅∆FED or ∆CAB≅∆FDE
• Invalid congruence statements; ∆ABC≅∆FED or
∆CAB≅∆DFE
Identifying Corresponding Congruent Parts
C
Z

A
B
X
Y
∆ABC is congruent to ∆XYZ
∆ABC is congruent to ∆XYZ
C
Z

A
B
X
Corresponding parts of these triangles are
congruent.
Y
∆ABC is congruent to ∆XYZ
C
Z

A
B
X
Corresponding parts of these triangles are
congruent.
Corresponding parts are angles and sides that
“match.”
Y
∆ABC is congruent to ∆XYZ
C
Z

A
B
X
Corresponding parts of these triangles are
congruent.
A

X
Y
∆ABC is congruent to ∆XYZ
C
Z

A
B
X
Corresponding parts of these triangles are
congruent.
B

Y
Y
∆ABC is congruent to ∆XYZ
C
Z

A
B
X
Corresponding parts of these triangles are
congruent.
C

Z
Y
∆ABC is congruent to ∆XYZ
C
Z

A
B
X
Y
Corresponding parts of these triangles are
congruent.
AB

XY
∆ABC is congruent to ∆XYZ
C
Z

A
B
X
Y
Corresponding parts of these triangles are
congruent.
BC

YZ
∆ABC is congruent to ∆XYZ
C
Z

A
B
X
Y
Corresponding parts of these triangles are
congruent.
AC

XZ
∆DEF is congruent to ∆QRS
F
Q

D
E
R
S
∆DEF is congruent to ∆QRS
F
Q

D
E
R
Corresponding parts of these triangles are
congruent.
S
∆DEF is congruent to ∆QRS
F
Q

D
E
R
Corresponding parts of these triangles are
congruent.
D

Q
S
∆DEF is congruent to ∆QRS
F
Q

D
E
R
Corresponding parts of these triangles are
congruent.
E

R
S
∆DEF is congruent to ∆QRS
F
Q

D
E
R
Corresponding parts of these triangles are
congruent.
F

S
S
∆DEF is congruent to ∆QRS
F
Q

D
E
R
Corresponding parts of these triangles are
congruent.
DE

QR
S
∆DEF is congruent to ∆QRS
F
Q

D
E
R
Corresponding parts of these triangles are
congruent.
DF

QS
S
∆DEF is congruent to ∆QRS
F
Q

D
E
R
Corresponding parts of these triangles are
congruent.
FE

SR
S
Example 1
Given that JKL  RST, list all
corresponding congruent parts.
SOLUTION
The order of the letters in the names of the triangles shows
which parts correspond.
Corresponding Angles
Corresponding Sides
∆JKL  ∆RST, so J  R.
∆JKL  ∆RST, so JK  RS.
∆JKL  ∆RST, so K  S.
∆JKL  ∆RST, so KL  ST.
∆JKL  ∆RST, so L  T.
∆JKL  ∆RST, so JL  RT.
Example 2
The two triangles are congruent.
a. Identify all corresponding
congruent parts.
b. Write a congruence statement.
SOLUTION
a. Corresponding Angles
Corresponding Sides
A  F
AB  FD
B  D
BC  DE
C  E
AC  FE
b. List the letters in the triangle names so that the corresponding
angles match. One possible congruence statement is ∆ABC  ∆FDE.
Your Turn:
Given STU  YXZ, list all corresponding
congruent parts.
ANSWER
ST  YX; TU  XZ;
SU  YZ; S  Y;
T  X; U  Z
Your Turn:
Which congruence statement is correct? Why?
A. JKL  MNP
B. JKL  NMP
C. JKL  NPM
ANSWER
B; This statement matches up the corresponding
vertices in order.
Practice Time! Your Turn
1) Are these shapes congruent?
Explain.
1) Are these shapes congruent?
Explain.

These shapes are congruent because they are
both parallelograms of equal size.
2) Are these shapes congruent?
Explain.
2) Are these shapes congruent?
Explain.
These shapes are not congruent because
they are different sizes.
3) Are these shapes congruent?
Explain.
3) Are these shapes congruent?
Explain.

These shapes are congruent because they are
the same size.
4) ∆BAD is congruent to ∆THE
Name all corresponding parts.
D
E

B
A
T
H
4) ∆BAD is congruent to ∆THE
Name all corresponding parts.
D
E

A
B
T
ANGLES
B
A
D



H
SIDES
T
BA
H
AD
E
DB



TH
HE
ET
5) ∆FGH is congruent to ∆JKL
Name all corresponding parts.
F
J

H
G
K
L
5) ∆FGH is congruent to ∆JKL
Name all corresponding parts.
F
J

G
H
K
ANGLES
F
H
G




L
SIDES
J
FG
L
GH
K
HF



JK
KL
LJ
6) ∆QRS is congruent to ∆BRX
Name all corresponding parts.
S
R
B
Q
X
6) ∆QRS is congruent to ∆BRX
Name all corresponding parts.
S
R
B
Q
X
ANGLES
Q
S
R



SIDES
B
QR
X
QS
R
SR



BR
BX
XR
7) ∆EFG is congruent to ∆FGH
Name all corresponding parts.
E
H
G
F
7) ∆EFG is congruent to ∆FGH
Name all corresponding parts.
E
H
G
F
ANGLES
E
F
G



SIDES
H
EF
F
EG
G
GF



HF
HG
GF
Stands for
Corresponding Parts of
Congruent Triangles are
Congruent
Definition CPCTC
• The bi-conditional phrase “if and only if” in the
congruent polygon definition means that both
the conditional and its converse are true.
• Therefore, definition of CPCTC is;
If the corresponding parts of two triangles are
congruent, then the two triangles are congruent.
AND
If two triangles are congruent, then the corresponding
parts of the two triangles are congruent.
CPCTC Practice
O
If CAT  DOG, then A  ___
CPCTC
because ________.
C
O
Add markings!
D
G
A
T
CPCTC Practice
If FJH  QRS, thenJH  RS
___
CPCTC
and F  Q
___ because _______.
If XYZ  ABC, then ZX  CA
___
and Y  B
___ because CPCTC
_______.
Example 3: Naming Congruent
Corresponding Parts
Given: ∆PQR  ∆STW
Identify all pairs of corresponding congruent parts.
Angles: P  S, Q  T, R  W
Sides: PQ  ST, QR  TW, PR  SW
Your Turn
If polygon LMNP  polygon EFGH, identify all pairs of
corresponding congruent parts.
Angles: L  E, M  F, N  G, P  H
Sides: LM  EF, MN  FG, NP  GH, LP  EH
Example 4
E
Use the two triangles at the right.
a. Identify all corresponding
congruent parts.
b.
Determine whether the triangles
are congruent. If they are congruent, write a
congruence statement.
F
D
SOLUTION
a. Corresponding Angles
Corresponding Sides
D  G
DE  GE
DEF  GEF
DF  GF
DFE  GFE
EF  EF
G
Example 4
E
F
D
G
b. All three sets of corresponding angles are
congruent and all three sets of corresponding sides
are congruent, so the two triangles are congruent. A
congruence statement is DEF  GEF.
Example 5
In the figure, HG || LK. Determine
whether the triangles are congruent.
If so, write a congruence statement.
SOLUTION
Start by labeling any information you can conclude from the figure.
You can list the following angles congruent.
HJG  KJL
H  K
G  L
Vertical angles are congruent.
Alternate Interior Angles Theorem
Alternate Interior Angles Theorem
The congruent sides are marked on the diagram, so
HJ  KJ, HG  KL, and JG  JL. Since all corresponding
parts are congruent, HJG  KJL.
Your Turn:
In the figure, XY || ZW. Determine
whether the two triangles are
congruent. If they are, write a
congruence statement.
ANSWER
yes; Sample answer: XVY  ZVW
Example 6
In the diagram, PQR  XYZ.
a. Find the length of XZ.
b. Find mQ.
SOLUTION
a. Because XZ  PR, you know that XZ = PR = 10.
b. Because Q  Y, you know that mQ = mY = 95°.
Your Turn:
Given ∆ABC  ∆DEF, find the length of DF and
mB.
ANSWER
3; 28°
Example 7A: Using Corresponding
Parts of Congruent Triangles
Given: ∆ABC  ∆DBC.
Find the value of x.
BCA and BCD are rt. s.
BCA  BCD
mBCA = mBCD
(2x – 16)° = 90°
2x = 106
x = 53
Def. of  lines.
Rt.   Thm.
Def. of  s
Substitute values for mBCA and
mBCD.
Add 16 to both sides.
Divide both sides by 2.
Example 7B: Using Corresponding
Parts of Congruent Triangles
Given: ∆ABC  ∆DBC.
Find mDBC.
mABC + mBCA + mA = 180°
mABC + 90 + 49.3 = 180
mABC + 139.3 = 180
∆ Sum Thm.
Substitute values for mBCA and mA.
Simplify.
mABC = 40.7
DBC  ABC
Subtract 139.3 from both sides.
Corr. s of  ∆s are  .
mDBC = mABC
Def. of  s.
mDBC  40.7°
Trans. Prop. of =
Your Turn
Given: ∆ABC  ∆DEF
Find the value of x.
AB  DE
Corr. sides of  ∆s are .
AB = DE
Def. of  parts.
2x – 2 = 6
2x = 8
x=4
Substitute values for AB and DE.
Add 2 to both sides.
Divide both sides by 2.
Your Turn
Given: ∆ABC  ∆DEF
Find mF.
mEFD + mDEF + mFDE = 180°
ABC  DEF
∆ Sum Thm.
Corr. s of  ∆ are .
mABC = mDEF
Def. of  s.
mDEF = 53°
Transitive Prop. of =.
mEFD + 53 + 90 = 180
mF + 143 = 180
mF = 37°
Substitute values for mDEF
and mFDE.
Simplify.
Subtract 143 from both sides.