Video Transcript
From the following figure, what can we conclude about a possible side-side-angle,
SSA, congruence criterion? Option (A) SSA is a valid congruence criterion. Option (B) SSA is not a valid congruence criterion. Option (C) SSA is a criterion that works sometimes. Or option (D) there is nothing we can conclude.
Let’s begin by looking at the two triangles in the figures we are given. We have two side lengths in the triangles which are congruent, since 𝐵𝐶 and 𝐵
prime 𝐶 prime are both given as five length units. Then, we have a second pair of side lengths which are congruent. 𝐴𝐵 and 𝐴 prime 𝐵 prime are both given as two length units. And we can also note that the measure of angle 𝐴𝐶𝐵 and the measure of angle 𝐴
prime 𝐶 prime 𝐵 prime are both equal to 16.1 degrees.
So, in the two triangles, we have two pairs of sides congruent and a pair of angles
congruent. Notice that we cannot say these two triangles are congruent by applying the
side-angle-side congruency criterion because the angle is not the included angle
between the two sides. And so, as the question asks, might there be a possible side-side-angle
criterion? Well, if we just looked at the information we had written, it might look like an
excellent start to a proof. But in fact, we need to have a closer look at the figures.
Remember that we can think informally of congruent figures as being the same shape
and size. But we can see that triangles 𝐴𝐵𝐶 and 𝐴 prime 𝐵 prime 𝐶 prime are clearly not
the same shape. So proving that two triangles have two pairs of sides congruent and a nonincluded
pair of angles congruent would not be enough to prove that the two triangles are
congruent, since we can see that by using these facts alone, two noncongruent
triangles have been drawn. And therefore, we can conclude that SSA is not a valid congruence criterion.
It might be easy to wonder why we didn’t select option (C). SSA is a criterion that works sometimes, because, after all, it would work
sometimes. And the best response to that is that we don’t have rules in mathematics that work
sometimes. We don’t say that sometimes the angle measures in a triangle add to 180 degrees
because we want to be sure that they always do. And we want the criteria that we use to prove triangles are congruent to always be
true. And if they only work sometimes, then they are not valid rules. So, SSA is not a valid congruence criterion.
