Video Transcript
In the given figure, triangle
𝐴𝐵𝐶 and 𝐵𝐶𝐷 have two equal sides and share one equal angle. Are triangles 𝐴𝐵𝐶 and 𝐵𝐶𝐷
congruent?
Let’s start by highlighting our
triangles. We have the slightly larger
triangle 𝐴𝐵𝐶 and then we have this slightly smaller triangle marked in orange,
𝐵𝐶𝐷. As we can clearly see, these are
different sizes and therefore would not be congruent. But as we indeed can see, we do
have some corresponding sides of equal length. And we do have a corresponding
angle of equal length. Perhaps we find some sort of
problem with the congruency rules. So let’s note down what we know
about each triangle and see if we can work out what’s happening.
We can see in our diagram that we
have two lengths marked as two units, the length 𝐴𝐵 and the length 𝐵𝐷, which we
could then write as congruent. The line 𝐵𝐶 of length four occurs
in both triangles. Finally, we can see that we have a
common angle. The angle 𝐴𝐶𝐵 would be equal to
the angle 𝐵𝐶𝐷. It might be tempting then to say
that we have a congruency because of a real SSA. But in fact, this is not a
congruency rule. Because as we can see from our
diagram, we could in fact create two noncongruent triangles using two corresponding
pairs of sides congruent and a pair of corresponding angles congruent.
We could have used the congruency
rule SAS if the angle was included between the two sides. But it’s not in this diagram. So therefore, we can happily say
that our two triangles are not congruent. They didn’t look congruent. And even in the case of a badly
drawn diagram, we can’t prove that they’re congruent.
