If two triangles have three congruent, corresponding angles, what additional information is needed to prove that the triangles are congruent?

Question: If two triangles have three congruent, corresponding angles, what additional information is needed to prove that the triangles are congruent? 

One of the most common questions in geometry is how to prove that two triangles are congruent. Congruent means that they have the same size and shape. There are several ways to prove that two triangles are congruent, depending on what information is given. In this blog post, we will focus on one of them: the angle-angle-angle (AAA) criterion.

The AAA criterion states that if two triangles have three congruent, corresponding angles, then they are similar. Similar means that they have the same shape, but not necessarily the same size. However, having three congruent angles is not enough to prove that two triangles are congruent. We need some additional information about the lengths of their sides.

To see why, consider the following example. Suppose we have two triangles ABC and DEF, such that angle A is congruent to angle D, angle B is congruent to angle E, and angle C is congruent to angle F. By the AAA criterion, we can conclude that triangle ABC is similar to triangle DEF. But are they congruent? Not necessarily. It is possible that triangle ABC is smaller or larger than triangle DEF, as shown in the figure below.


As you can see, having three congruent angles does not guarantee that the triangles have the same size. Therefore, we need some additional information about the lengths of their sides to prove that they are congruent. One way to do that is to use the side-side-side (SSS) criterion, which states that if two triangles have three congruent, corresponding sides, then they are congruent. Another way is to use the side-angle-side (SAS) criterion, which states that if two triangles have two congruent, corresponding sides and the included angle between them is also congruent, then they are congruent.

In summary, if two triangles have three congruent, corresponding angles, we can only say that they are similar, not congruent. To prove that they are congruent, we need some additional information about the lengths of their sides. We can use either the SSS criterion or the SAS criterion to do that.

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