Geometry: Congruent Triangles
Congruent triangles are triangles that
have the same size and shape. This means that the
corresponding sides are equal and the corresponding angles are equal.

In the above diagrams, the corresponding
sides are a and d; b and e ; c and f.
The corresponding angles are x
and s; y
and t; z
and u.
We can tell whether two triangles are congruent
without testing all the sides and all the angles of the two triangles.
There are four rules to check for congruent triangles. They are
called the SSS rule, SAS rule,
ASA rule and AAS rule. As
long as one of the rules is true, it is sufficient to prove that
the two triangles are congruent.
If three sides of one triangle are equal
to three sides of another triangle, then the triangles are congruent.
(Hence the name of this rule: Side-Side-Side, SSS)
If two sides and the included angle (Side-Angle-Side,
SAS) of one triangle are equal to two sides and included angle of
another triangle, then the triangles are congruent. An included
angle is the angle formed by the two given sides.

Included Angle
Non-included
angle
If two angles and the included side of
one triangle (Angle-Side-Angle, ASA) are equal to two angles and
included side of another triangle, then the triangles are congruent.
An included side is the side between the two given angles.
If two angles and a non-included side of
one triangle (Angle-Angle-Side, AAS) are equal to two angles and
a non-included side of another triangle, then the triangles are
congruent. (This rule may sometimes be refered to as SAA).
For the ASA rule the given side must be
included and for AAS rule the side given must not be included. The
trick is we must use the same rule for both the triangles that we
are comparing.

Compare AAS with AAS

Compare ASA with ASA
Compare AAS with ASA
Example 1:

Which of the following conditions would
be sufficient for the above triangles to be congruent?
a) a = e, x = u, c = f
b) a = e, y = s, z = t
c) x = u, y = t, z = s
d) a = f, y = t, z = s
Solution for
a):
Step 1: a = e gives the S
x = u gives the A
c = f gives the S
Step 2: Beware! x and u
are not the included angles. This is not SAS
but ASS which is not one of the rules. Note that you cannot
compare donkeys with triangles!
Answer:
a = e, x = u, c = f is not sufficient for the above
triangles to be congruent.
Solution for
b):
Step 1: a = e gives the S
y = s gives the A
z = t gives the A
Step 2: a and e are
non-included sides. Follows the AAS rule.
Answer:
a = e, y = s, z = t is sufficient show that the above are
congruent triangles.
Solution for
c):
Step 1: x = u gives the A
y = t gives the A
z = s gives the A
Step 2: AAA is not one of
the rules.
Answer:
x = u, y = t, z = s is not sufficient for the above
triangles to be congruent.
Solution for
d):
Step 1: a, y, z follows AAS
(non-included side)
f ,t, s follows the ASA (included side)
Step 2: Comparing AAS with ASA is
not allowed
Answer:
a = f, y = t, z = s is not sufficient to show that
the above are congruent triangles.
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