Video Transcript
In the given figure, line segment 𝐷𝐸 is constructed on the triangle 𝐴𝐵𝐶 parallel to line segment 𝐵𝐶. What can we conclude about the measures of angle 𝐴𝐷𝐸 and 𝐴𝐵𝐶? Using the AA criterion, what can we conclude about triangles 𝐴𝐷𝐸 and 𝐴𝐵𝐶?
The first thing that we’re told about this figure is that line segment 𝐷𝐸 is parallel to line segment 𝐵𝐶. There are also two triangles, which we’ll be considering more in the second part of the question. We have the smaller triangle, triangle 𝐴𝐷𝐸, and then we have the larger triangle, triangle 𝐴𝐵𝐶. Let’s have a look at the first part of this question then, which asks us to consider angle 𝐴𝐷𝐸 which is here on the smaller triangle with angle 𝐴𝐵𝐶 which is part of the larger triangle. Let’s remember that we have two parallel lines, the line segment 𝐷𝐸 and the line segment 𝐵𝐶. This line 𝐴𝐵 can then be considered as a transversal of these parallel lines.
Angle 𝐴𝐷𝐸 and angle 𝐴𝐵𝐶 are therefore corresponding angles. And as they’re corresponding, this means that they’re equal. We could give the answer then for the first part of the question that the measure of angle 𝐴𝐷𝐸 is equal to the measure of angle 𝐴𝐵𝐶.
In the second part of the question, we’re using the AA criterion to conclude something about these two triangles. When we hear the AA criterion, that means that we’ll be thinking about similar triangles. The AA criterion applies when we have two pairs of corresponding angles equal, which indicates that we have two similar triangles. Although, in the first part of this question, we did find two equal angles, it’s important not to get confused into thinking that this is two angles for the AA rule. In fact, it would just be one pair of equal angles, in other words, just one 𝐴 out of the AA rule. In order to prove that triangles 𝐴𝐷𝐸 and 𝐴𝐵𝐶 are similar, we’ll need to find another pair of equal corresponding angles.
Let’s have look at this angle, angle 𝐷𝐴𝐸, in triangle 𝐴𝐷𝐸. We might notice that it’s exactly the same angle as angle 𝐵𝐴𝐶 in triangle 𝐴𝐵𝐶. These two triangles share this common angle at 𝐴, so these two angles are equal. We have therefore shown as we saw in the first part of the question that we have one pair of corresponding angles equal and then we have a second pair of corresponding angles equal. We can, therefore, apply the AA criterion. So what can we conclude about triangles 𝐴𝐷𝐸 and 𝐴𝐵𝐶? Well, the answer is they are similar. Of course, similar triangles have three pairs of corresponding angles congruent.
And although we only need to show that two pairs of these are congruent to demonstrate similarity, we could also have shown that the third pair of angles are equal. Angle 𝐴𝐸𝐷 is equal to angle 𝐴𝐶𝐵 because, once again, like in the first part of the question, we have a pair of parallel lines and the transversal 𝐴𝐶. So therefore, these two angles would be corresponding. Any two pairs out of the three pairs of angles in the triangle would be sufficient to fulfill the AA criterion and therefore prove these triangles are similar.
