In the given figure, 𝐴𝐵 and 𝐷𝐸 are parallel. Using the AA criterion, what can we say about triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐶?
So we’ve been asked about a pair of triangles and the logical responses here are perhaps that they are similar triangles or perhaps they’re congruent triangles or there may be another type of relationship.
The question tells us to use the AA criterion, where AA means angle-angle. So we’ve been asked to look at the angles of these two triangles. We’ll start by writing down some statements about their angles. The angle marked in orange is common to both triangles. It’s angle 𝐴𝐶𝐵 in the larger triangle and angle 𝐷𝐶𝐸 in the smaller.
So we have the statement that angle 𝐴𝐶𝐵 is congruent to angle 𝐷𝐶𝐸. And the reasoning for this is that it’s common or shared. This is our first statement about the angles in the two triangles. So we write an A in brackets next to it to indicate it’s a statement about angles. Next, let’s consider the pair of angles marked in green.
The question tells us that the lines 𝐴𝐵 and 𝐷𝐸 are parallel, which means these two angles are in fact corresponding angles. You may be able to see this more clearly if you imagine extending the pair of parallel lines. This pair of green angles are both in the same relative position where the transversal meets the parallel lines.
They are to the right of the parallel lines and below the transversal. So we have a second statement about the angles in these two triangles. Angle 𝐵𝐴𝐶 is congruent to angle 𝐸𝐷𝐶. The reasoning, remember, is that they’re corresponding angles. And again we use an A to denote a fact about angles.
So we’ve shown that two of the angles in these triangles are equal. This is what is meant by the AA criterion. And it is sufficient to prove that these two triangles are similar. Could they be congruent? Well, no because if you look at the two triangles, they’re clearly of different sizes. And in order for them to be congruent, they not only need to have the same angles, but also need to have the same lengths.
Therefore, we can conclude that the two triangles in the question are similar. Now just as an aside, you may be thinking there are three angles in a triangle. So why do we only need to demonstrate the two of them are equal in order to conclude that the triangles are similar?
Well, this is because the angle sum in a triangle is fixed at 180 degrees. So if you’ve demonstrated that two angles are the same, the third also must be. This is why it is sufficient to use an angle-angle criterion to show that two triangles are similar, not angle-angle-angle as the third angle follows by default.