In the figure, 𝐴𝐵 is parallel to
𝐷𝐸. Which congruence criterion could be
used to prove the two triangles are congruent?
So, in this question we have two
triangles, triangle 𝐴𝐵𝐶 and triangle 𝐶𝐷𝐸. We need to check if they are
congruent, which means they’re exactly the same shape and the same size. Let’s remind ourselves of the
criteria which must be met if we want to show that two triangles are congruent. The first option is SSS which
stands for side, side, side. In this case, we would need to show
that there are corresponding sides which are equal in each triangle.
The second option is SAS, which is
side, angle, side, where the angle has to be the included angle between the two
sides. The third possibility is angle,
side, angle where the side is the included side between the two given angles. Alternatively, we could have two
angles and a side. Or finally, in our option for right
triangles only, if we have the hypotenuse and a leg, that’s a side of the triangle
corresponding, then that would show congruence.
So, let’s take a look at our
diagram and see what we could establish about the properties of these two
triangles. We could begin by marking on our
two parallel lines 𝐴𝐵 and 𝐷𝐸. Since we have two parallel lines,
this means that the measure of angle 𝐵𝐴𝐶 is equal to the measure of angle 𝐶𝐸𝐷
since these are alternate interior angles. We also have another set of
alternate interior angles. So, we could say that the measure
of angle 𝐴𝐵𝐶 is equal to the measure of angle 𝐶𝐷𝐸. And for the third angle, we can say
that the measure of angle 𝐴𝐶𝐵 is equal to the measure of angle 𝐷𝐶𝐸 because
these are vertical or opposite angles.
So if we take a look at our
diagram, we can’t say anything for definite about the lengths of the sides. We weren’t given any measurements
and we can’t measure with a ruler because we weren’t told that the diagram was drawn
to scale. So, if we look at our congruency
criteria, we can’t use the first four options because they involve the side
lengths. We also can’t use the final
congruency criterion because we don’t know the length of the sides or hypotenuse,
and we don’t know if any of the angles is a right angle.
The only thing we have been able to
show in our diagram is that we have three corresponding angles equal to each
other. But this is not sufficient to show
congruence. If we have two triangles, which
have three corresponding angles equal to each other, this would only show that
they’re similar but not necessarily congruent. So, since we haven’t been able to
demonstrate any of the congruency criteria have been fulfilled, then the answer is
there’s not enough information to prove congruence.