Is AAA (Angle-Angle-Angle) a Congruence Rule?

(Video) Congruent Triangles AAA

by Alton Price

We know that AAS, ASA are both valid congruence rules. Both rules requires two pairs of congruent angles. However, are 3 congruent angles enough to prove the congruence of two triangles? $\\$ Here is a video demonstrating why AAA is NOT a valid congruence rule.

As you can see in the video, triangles that have 3 pairs of congruent angles do not necessarily have the same size. $\\$ AAA (Angle-Angle-Angle) is not a congruence rule!

We have already explored the ASA and SAS congruence rules. Now we are going to consider when two triangles have all three of the same angles!

Consider these two triangles with the same angles

If you look closely enough, the red triangle in figure 1 has side lengths shorter than the black triangle in figure 1! But they seem to be similar; it almost looks like the ratio of their side lengths are the same. Side $AB$ seems to be twice the length of side $DE$, side $BC$ twice the length of side $EF$, and side $AC$ seems to be twice the length of $DF$.

Are triangles with the same three angles always similar? Consider the following example:

Even though the triangles are oriented in different directions, they are still similar! This time, the length of $BC$ is twice $DE$ rather than $EF$ and the length $AB$ is twice $EF$ rather than $DE$!

The takeaway is this: congruent triangles have the same angles but just because two triangles have the same angles does not mean that they are congruent but they must be similar.