Congruence Shortcuts Notes

1. •Two triangles are congruent if one can
be placed on top of the other for a
perfect match (they have the same size
and shape).
•In the figure, is congruent to
In symbols:
•Just as with similar triangles, it is
important to get the letters in the correct
order. For example, since A and D
come first, we are saying that when the
triangles are made to coincide, A and D
will coincide.
A
B C
D
E F
ABCD
.DEFD .ABC DEFD @D
Definition of Congruent Triangles

2. CPCTC
ABC DEFD @D
ABDEAD
ACDFBE
BCEF CF
@ @
@ @
@ @
•Corresponding parts of congruent
triangles are congruent (CPCTC).
•What this means is that if
then:
•Other corresponding “parts” (like
medians) are also congruent.
A
B C
D
E F

3. •To prove that two triangles are congruent it is
only necessary to show that some
corresponding parts are congruent.
•For example, suppose that in and in
that
•Then intuition tells us that the remaining sides
must be congruent, and…
•The triangles themselves must be congruent.
ABCD
DEFD
ABDE@
and ACDF@
and AD@
A
D
CB
FE
Proving Triangles Congruent

4. SAS
•In two triangles, if one pair of sides are
congruent, another pair of sides are congruent,
and the pair of angles in between the pairs of
congruent sides are congruent, then the
triangles are congruent.
•For example, in the figure, if the
corresponding parts are congruent as marked,
then
•We cite “Side-Angle-Side (SAS)” as the
reason these triangles are congruent.
ABCDEFD @D
A
B C
D
E F

5. •In two triangles, if all three pairs of
corresponding sides are congruent then the
triangles are congruent.
•For example, in the figure, if the
corresponding sides are congruent as
marked, then
•We cite “side-side-side (SSS)” as the reason
why these triangles are congruent.
A
B
C
D
E
F
ABCDEFD @D
SSS

6. ASA
•In two triangles, if one pair of angles are
congruent, another pair of angles are
congruent, and the pair of sides in between
the pairs of congruent angles are congruent,
then the triangles are congruent.
•For example, in the figure, if the
corresponding parts are congruent as
marked, then
•We cite “angle-side-angle (ASA)” as the
reason the triangles are congruent.
ABCDEFD @D
A B
C
D E
F

7. AAS
•In two triangles, if one pair of angles are
congruent, another pair of angles are
congruent, and a pair of sides not between the
two angles are congruent, then the triangles
are congruent.
•For example, in the figure, if the
corresponding parts are congruent as marked,
then
•We cite “angle-angle-side (AAS)” as the
reason the triangles are congruent.
ABCDEFD @D
A B
C
D E
F