Video Transcript
Which congruence criterion can be
used directly to prove that triangles 𝐴𝐵𝐶 and 𝐴𝐷𝐶 in the given figure are
congruent? Option (A) ASA, option (B) SAS,
option (C) SSS, or option (D) RHS.
Let’s begin this problem by
identifying the two triangles. Triangle 𝐴𝐷𝐶 is on the left, and
triangle 𝐴𝐵𝐶 is on the right. And we are asked to determine which
of the congruency criteria we could use to prove that these two triangles are
congruent. The first thing we might observe is
that both these triangles have a right angle, which means they are both right
triangles.
And there is one congruency
criterion which applies in right triangles. It is the RHS criterion, which
means right angle-hypotenuse-side. So have we got enough information
to apply this criterion here? We can start by noting down the
information about the right angles, as we have the two angles 𝐴𝐷𝐶 and 𝐴𝐵𝐶,
which both have a measure of 90 degrees.
Next in the criterion would be the
hypotenuse of the right triangle. If it’s difficult to work out which
side is the hypotenuse, it’s always helpful to remember that in a right triangle,
the hypotenuse is always the side opposite the right angle. Since this line segment 𝐴𝐶, which
is the hypotenuse of both triangles, is a common side, then we know that this length
is congruent in each of the triangles 𝐴𝐵𝐶 and 𝐴𝐷𝐶. Finally, we can note that the line
segments 𝐶𝐷 and 𝐶𝐵 are marked as congruent. So, we have another pair of
congruent sides in each triangle.
And so, we have demonstrated that
each triangle has a right angle, the hypotenuse in each is congruent, and there is
another pair of sides congruent. Therefore, we can prove that
triangles 𝐴𝐵𝐶 and 𝐴𝐷𝐶 are congruent using the RHS criterion, which was the
answer given in option (D).
As an aside, it is common that when
we are dealing with congruent triangles, there may be more than one congruence
criterion which we could use. So, let’s look at the other
options.
We can exclude the angle-side-angle
criterion given in option (A), as there is only information about one angle in the
diagram, which was the right angle. We can also exclude option (B), the
side-angle-side criterion, as the 90-degree angle we were given was not included
between the two sides whose congruency we could establish. And finally, the SSS rule cannot be
applied directly as we weren’t given any information about the congruency of all
three pairs of sides.
Therefore, the RHS criterion is the
only one we could directly apply from these options.
