AAA/AAS/ASA/SAS/SSS Many arguments and proofs
presented in the study of
GEOMETRY
rely on identify-
ing similar triangles. The
SECANT
theorem, for
instance, illustrates this. Fortunately, there are a num-
ber of geometric tests useful for determining whether
or not two different triangles are similar or congruent.
The names for these rules are acronyms, with the let-
ter Astanding for the word angle, and the letter Sfor
the word side. We list the rules here with an indica-
tion of their proofs making use of the
LAW OF SINES
and the
LAW OF COSINES
:
a. The AAA rule: If the three interior angles of one tri-
angle match the three interior angles of a second tri-
angle, then the two triangles are similar.
The law of sines ensures that pairs of corresponding
sides of the triangles have lengths in the same ratio.
Also note, as the sum of the interior angles of any trian-
gle is 180°, one need only check that two corresponding
pairs of interior angles from the triangles match.
b. The AAS and ASA rules: If two interior angles and
one side-length of one triangle match corresponding
interior angles and side-length of a second triangle,
then the two triangles are congruent.
By the AAA rule the two triangles are similar. Since a
pair of corresponding side-lengths match, the two trian-
gles are similar with scale factor one, and are hence con-
gruent. (Note that any two right triangles sharing a
common hypotenuse and containing a common acute
angle are congruent: all three interior angles match, and
the AAS and ASA rules apply. This is sometimes called
the “HA congruence criterion” for right triangles.)
c. The SAS rule: If two triangles have two sides of
matching lengths with matching included angle, then
the two triangles are congruent.
The law of cosines ensures that the third side-lengths of
each triangle are the same, and that all remaining angles
in the triangles match. By the AAS and ASA rules, the
triangles are thus congruent. As an application of this
rule, we prove E
UCLID
’s isosceles triangle theorem:
The base angles of an isosceles triangle are
equal.
Suppose ABC is a triangle with sides AB and
AC equal in length. Think of this triangle as
representing two triangles: one that reads BAC
and the other as CAB. These two triangles
have two matching side-lengths with matching
included angles, and so, by the SAS rule, are
congruent. In particular, all corresponding
angles are equal. Thus the angle at vertex Bof
the first triangle has the same measure as the
corresponding angle of the second triangle,
namely, the angle at vertex C.
This result appears as Proposition 5 of Book I of
Euclid’s famous work T
HE
E
LEMENTS
.
d. The SSS rule: If the three side-lengths of one triangle
match the three side-lengths of a second triangle,
then the two triangles are congruent.
1
A