4.2: Congruent Triangles
 Two figures are congruent if they are
size
shape
the same ______
and same _______.
Segments are congruent if they
have the same length.
Angles are congruent if they have the
same measure.
Congruent Triangles
 Two triangles are congruent if each
part of one triangle is congruent to a
corresponding part of the other (both
angles and sides… 6 altogether).
A
X
ABC  ZXY
C
Y
B
Z
Congruent Triangles
 We want to prove that two triangles
are congruent. Luckily, we don’t have
to go through and prove that every
set of corresponding parts are
congruent… we have some shortcuts…
yippee!
Proving Triangles are Congruent
 SSS postulate (side-side-side):
 If 3 sides of one triangle are congruent
to 3 sides of another, then the triangles
are congruent.
A
X

C
B
Y
(by SSS)
Z
Ex. 1
 Given: JK  JM ; KL  ML
 Prove: JKL  JML
K
J
L
M
Ex. 1
 Statement
Reason
1) JK = JM, KL = MK
1) Given
2) JL = JL
2) Reflexive Property
3)JKL  JML
3) SSS
Proving Triangles are Congruent
 SAS postulate (side-angle-side):
 If 2 sides and the included angle of
both triangles are congruent, then
the triangles are congruent.
A
X

C
B
Y
(by SAS)
Z
Proving Triangles are Congruent
 ASA postulate (angle-side-angle):
 If 2 angles and the included side of
both triangles are congruent, then
the triangles are congruent.
A
X

C
B
Y
(by ASA)
Z
Ex. 2:
 Given: JK  ML; JK || ML
 Prove: JKL  MLJ
J
K
1
M
2
L
Ex. 2
 Statement
Reason
1) JK = ML, JK || ML
1) Given
2) JL = JL
2) Reflexive Property
3)
1  2
4)JKL  MLJ
3) Alternate Interior Angles Theorem
4) SAS
Ex. 3:
 Given: VW  ZY ; V  Z ;VW  WY ; ZY  WY
 Prove: VWX  ZYX
V
W
Z
X
Y
Ex. 3
 Statement
1) VW = ZY,
ZY  WY
2)
3)
4)
5)
V  Z
VW  WY
W and Y are right angles
mW  90; mY  90
W  Y
VWX  ZYX
Reason
1) Given
2) Definition of Perpendicular lines
3) Definition of Right Angle
4) Transitive Property of Equality
5) ASA