Question Video: Using the Angle-Side-Angle Triangle Congruence Criterion to Establish Congruence Mathematics • First Year of Preparatory School

Video Transcript

Which congruence criteria can be used to prove that the two triangles in the given figure are congruent?

In this question, we need to determine how we can show these triangles are congruent, which means that their corresponding sides are congruent and corresponding angles are congruent. Let’s take a look at the measurements that we’re given on the figure. We can see that we have a pair of lines, which are both given as 2.57 length units, which means that 𝐴𝐵 must be equal to 𝐴 prime 𝐵 prime. We also have two pairs of congruent angle measures. The measure of angle 𝐴𝐵𝐶 is equal to the measure of 𝐴 prime 𝐵 prime 𝐶 prime because they both measure 65.03 degrees. And the measure of angle 𝐴𝐶𝐵 is equal to the measure of angle 𝐴 prime 𝐶 prime 𝐵 prime, as these are both equal to 58.55 degrees.

Therefore, we have found that there are a pair of congruent sides and two pairs of corresponding congruent angles. Now, there is a congruency criterion ASA which relates two angles and a side. It states that two triangles are congruent if they have two angles congruent and the included side congruent. However, in these triangles, the side is not the included side because it doesn’t lie between the two angles. This means that we can’t immediately apply the ASA criterion.

But as we are given two angles in the triangle, we could calculate the third angle in each triangle by using the fact that the angle measures in a triangle sum to 180 degrees. In triangle 𝐴 prime 𝐵 prime 𝐶 prime, we can calculate the third angle as 180 degrees subtract 58.55 degrees plus 65.03 degrees. This would give us that the measure of angle 𝐵 prime 𝐴 prime 𝐶 prime is 56.42 degrees. In triangle 𝐴𝐵𝐶, the third angle will also be equal to 180 degrees subtract 58.55 degrees plus 65.03 degrees. And as these are the same values, then we know that the measure of angle 𝐵𝐴𝐶 will also be 56.42 degrees.

So now if we compare the triangles, we have two angles of 56.42 degrees and another pair of angles of 65.03 degrees. The side of 2.57 length units is the included side between these two angles. We could now apply the ASA congruence criterion to show that these two triangles are congruent. For the answer then, we would need to give angle side angle or ASA.

Note, however, that if we were answering a question like this in an exam, we would need to very clearly show that we had calculated the third angle in the triangle. Without knowing this third angle, we could not have applied the ASA congruence criterion.