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.
