Triangle congruence
Triangles Congruence
Review angles in triangles
1. Determine the measure of the angle.
a) mA = ________
b) mC = ________
B
c) mC = ________
B
d) mA = ________
B
B
60°
48°
99°
118°
C
65°
41°
C
A
A
A
A
C
C
2. Determine the measure of the angles.
a) m1 = _______ m2 =
______
1
b) m1 = _______ m2 = ______ c) m1 = _______ m2 =
______
m3 = _______
m3 = _______
2
64°
27°
76°
1
2
1
3
2
68°
3
3. Determine the missing information.
a) x = _______ mA =
_________
c) x = _______ mA =
_________
B
B
B
68°
4x
2x
3x - 3
A
b) x = _______ mB =
_________
3x
2x
A
C
2x
3x + 5
C
A
C
Triangle congruence
5. Determine the missing information.
a) m1 = ______ m3 =______ b) m1 = ______ m3 = ______
B
B
c) m1 = ______ m3 = ______
98°
40°
1
3
49°
18°
124°
3
A
1
165°
3
1
C
C
A
6. Find the value of x
a) x = ____________
b) x = ____________
c) x = ____________
2x - 2
x
3x - 17
x
x
x + 40
x+5
d) x = ____________
e) x = ____________
2x - 5
f) x = ____________
E
47°
B
42°
x
48°
64°
81°
x
A
x
F
g) x = ____________
52°
h) x = ____________
x
82°
C
D
i) x = ____________
121°
56°
59°
x
x
65°
82°
Triangle congruence
Congruence of triangles is defined by:
When triangles are congruent and one triangle is placed on top of the other, the sides and
angles that coincide (are in the same positions) are called corresponding parts.
Two triangles are congruent if and only if one can be mapped onto the other
by one or more rigid motions.
Example:
When two triangles are congruent, there are
6 facts that are true about the triangles:


the triangles have 3 sets of congruent
(of equal length) sides and
the triangles have 3 sets of congruent
(of equal measure) angles.
NOTE: The corresponding
congruent sides are marked
with small straight line
segments called hash marks.
The corresponding congruent
angles are marked with arcs.
The 6 facts for our congruent triangles example:
Triangle congruence
Establishing Triangle Congruence
Methods for Proving (Showing) Triangles to be Congruent
SSS
If three sides of one triangle are congruent to three sides of another triangle,
the triangles are congruent.
(For this method, the sum of the lengths of any two sides must be greater than the length of the third side,
to guarantee a triangle exists.)
A
E
B
B
H
C
R
R
W
A
SAS
T
C
D
J
If two sides and the included angle of one triangle are congruent to the
corresponding parts of another triangle, the triangles are congruent. (The included
angle is the angle formed by the sides being used.)
H
A
o
Y
o
G
B
K
A
C
B
o
F o
ASA
U
M
o
M
o
H
C
If two angles and the included side of one triangle are congruent to the
corresponding parts of another triangle, the triangles are congruent. (The included
side is the side between the angles being used. It is the side where the rays of the angles would overlap.)
E
o
E
o
T
x
T
x
G
G
x
E
o
Y
x
o
x
G
x
o
L
K
D
B
AAS
T
o
U
If two angles and the non-included side of one triangle are congruent to the
corresponding parts of another triangle, the triangles are congruent. (The nonincluded side can be either of the two sides that are not between the two angles being used.)
E
o
T
x
G
x
K
HL
Right
Triangles
Only
E
o
U
o
L
T
*
x
E
U
U
*
*
G
x
o
L
T
K
If the hypotenuse and leg of one right triangle are
congruent to the corresponding parts of another right
triangle, the right triangles are congruent. (Either leg of the
right triangle may be used as long as the corresponding legs are used.)
*
x
L
G
x
K
Triangle congruence
BE CAREFUL!!!
Only the combinations
listed above will give
congruent triangles.
So, why do other combinations not work?
Methods that DO
AAA
NOT Prove Triangles to be Congruent
You can easily draw 2 equilateral
triangles that are the same shape
same SHAPE (similar), but does NOT work to also
but are not congruent (the same
show they are the same size, thus congruent!
size).
AAA works fine to show that triangles are the
Consider the example at the right.
SSA
This is NOT a universal method to prove triangles congruent because it cannot
guarantee that one unique triangle will be drawn!!
or
ASS
The combination of SSA (or ASS) creates a unique triangle ONLY when working in a
right triangle with the hypotenuse and a leg. This application is given the name HL
(Hypotenuse-Leg) for Right Triangles to avoid confusion. You should not list SSA
(or ASS) as a reason when writing a proof.
Once you prove your triangles are congruent, the "left-over" pieces that were not used in your
method of proof, are also congruent. Remember, congruent triangles have 6 sets of congruent
pieces. We now have a "follow-up" theorem to be used AFTER the triangles are known to be
congruent:
Theorem: (CPCTC) Corresponding parts of congruent triangles
are congruent
Triangle congruence
Formal Proofs
When attempting to prove triangles congruent, it is important to satisfy all of the conditions of
the congruent triangle method you are using. In each problem below, examine the diagram
and the GIVEN information.
 Determine the method needed to prove the triangles congruent.
(ASA, SAS, AAS, SSS, or HL for right triangles only
 Each of the three components needed to support the chosen method appear to the left of
their corresponding Statement.
 Decide what Reasons can be used to support your decisions.
1.
Method
S
A
S
Statement
1.
2.
3.
4.
Reason
Given
Given
Given
SAS
2
Method Statement
S
1.
A
S
2.
3.
4.
Reason
Midpoint of a segment divides a
segment equally
Vertical angles congruent
Given
SAS
Triangle congruence
3
Method
A
A
S
Statement
1.
2.
3.
4.
Reason
Given
Given
Reflexive property – shared
side
AAS
4
Method Statement
1.
A
S
2.
A
3.
4.
Reason
Given
Midpoint of a segment divides a
segment equally
Given
ASA
Triangle congruence
5
Method Statement
S
1.
S
2.
3.
S
4.
5.
Reason
Given
Bisector intersects at the
midpoint
Midpoint of a segment
divides a segment equally
Midpoint of a segment
divides a segment equally
SSS
6
Method Statement
1.
2.
Right
triangle
H
3.
L
4.
5.

Reason
Given
Right triangle contains a
right angle
Given
Reflexive property – shared
side
HL
Triangle congruence
1. Prove the following relationships.
a) GIVEN:
B   E &
E
VB  DC
PROVE:
B
C
EAD  BAC
STATEMENT
REASON
c) GIVEN:
E
A
D
ACB  ECD
STATEMENT
REASON
A
E
AE  BE &
B
B
D
D
BVC  CDB
STATEMENT
REASON
B
AB  CD
C
A
D
PROVE:
ABD  CDB
STATEMENT
f) GIVEN:
T is the midpoint of
REASON
G
ME
DE  CE
PROVE:
PROVE:
AD  CB &
C
PROVE:
e) GIVEN:
C
d) GIVEN:
B
B  D &
V
VC  DB &
A
CB  DE
BC  DC
b) GIVEN:
D
C
AED  BEC
J
& T is the midpoint of
GJ
E
T
M
PROVE:
STATEMENT
REASON
MGT  EJT
STATEMENT
REASON
Triangle congruence
2. Prove the following relationships.
a) GIVEN:
BC  DC &
AC  EC
PROVE:
A  E
STATEMENT
b) GIVEN:
B
E
D  C &
C
E
A
D
REASON
Do Worksheets







A
Recognizing congruent triangles
Congruent Triangles
Congruent Triangles MC practice
Completing triangle proofs
Proof outlines
Congruent Triangle proofs
Scrambled Proofs
B
DE  CE
PROVE:
 AD  BC
STATEMENT
D
C
REASON