The figure shows two triangles, 𝐴𝐵𝐶 and 𝐴 prime 𝐵 prime 𝐶 prime. Work out the measure of angle 𝐴𝐵𝐶. What does the AA criterion tell us about these two triangles?
If we look at the diagram, we can see that there are two triangles. 𝐴𝐵𝐶 is the smaller triangle, and 𝐴 prime 𝐵 prime 𝐶 prime is the larger triangle. Let’s begin by looking at the first question to work out the measurement of angle 𝐴𝐵𝐶, and it’s down here as part of the smaller triangle 𝐴𝐵𝐶. We should remember that the angles in a triangle add up to 180 degrees, and we’re given the other two angles in this triangle. We can therefore write that angle 𝐴𝐵𝐶, that’s the angle we wish to find out, is equal to 180 degrees subtract the sum of the other two angles. That’s angle 𝐵𝐴𝐶 and angle 𝐴𝐶𝐵. We can then simply fill in the values of these two known angles. 111 plus 22.7 gives us 133.7. And then, subtracting this from 180 degrees gives us 46.3 degrees.
We have therefore answered the first part of this question. Angle 𝐴𝐵𝐶 is 46.3 degrees.
If we take a look at the figure, we might notice that we have another angle, which is also 46.3 degrees. It’s this angle 𝐴 prime 𝐵 prime 𝐶 prime. Indeed, we also have another pair of equal angles. We have two angles which are both marked as 111 degrees. This will help us in the second part of the question. This question asks, what does the AA criterion tell us about these two triangles? The AA criterion is something that is used to prove that two triangles are similar. We remember that similar triangles have corresponding angles equal and corresponding sides in proportion. We should begin this question by considering what the AA criterion actually is and if indeed it applies to these two triangles.
AA just means angle-angle. So we need to check if there are two pairs of angles equal. Well, we know that there are because we’ve already established that the two pink angles, angle 𝐴𝐵𝐶 and angle 𝐴 prime 𝐵 prime 𝐶 prime, are equal. And we also have angle 𝐵𝐴𝐶 is equal to angle 𝐵 prime 𝐴 prime 𝐶 prime as these two angles are both 111 degrees. In order to write a good statement to answer the second question, we need to demonstrate that we understand what AA means and the fact that it shows that the triangles are similar. Therefore, we could write that as both triangles share two angles of equal measures, they must be similar.