Video Transcript
Triangles 𝐴𝐷𝐸 and 𝐴𝐵𝐶 in the
given figure are similar. What, if anything, must be true of
the lines 𝐷𝐸 and 𝐵𝐶? Option (A) they are parallel, or
option (B) they are perpendicular.
In this question, we are told that
there are two similar triangles. They are triangle 𝐴𝐷𝐸, which is
the smaller triangle, and triangle 𝐴𝐵𝐶, which is the larger triangle. We can recall that similar
triangles have corresponding angles congruent and corresponding sides in
proportion.
Now, we are asked what must be true
of the two lines 𝐷𝐸 and 𝐵𝐶. But we are aren’t given any
information about the lengths of any sides in this figure. So, let’s see what we can work out
by using the angle properties of these similar triangles. Since the corresponding angles are
congruent, we know that the measure of angle 𝐴𝐷𝐸 is equal to the measure of angle
𝐴𝐵𝐶. And, in the same way, the
corresponding angles 𝐴𝐸𝐷 and 𝐴𝐶𝐵 must be of equal measure.
The third pair of angles in each
triangle is the common angle at vertex 𝐴, which we could refer to as angle 𝐷𝐴𝐸
in triangle 𝐴𝐷𝐸 and angle 𝐵𝐴𝐶 in triangle 𝐴𝐵𝐶. But we can consider the information
from the first pair of angles. These angles are constructed
between the lines 𝐷𝐸 and 𝐵𝐶 and the line 𝐴𝐵. And we know that these angles are
congruent.
We know that if we have a pair of
parallel lines and a transversal, then the corresponding angles are congruent. And remember that the converse of
this is also true. That is, if corresponding angles in
a transversal of two lines are congruent, then the lines are parallel. And this is the situation that we
have here. The two lines are 𝐷𝐸 and
𝐵𝐶. The transversal is the line
𝐴𝐵. And corresponding angles are
congruent. Therefore, the lines 𝐷𝐸 and 𝐵𝐶
are parallel. So, the answer to the question is
that given in option (A). We can say that the lines 𝐷𝐸 and
𝐵𝐶 are parallel.
It’s worth noting that we could
have proved the same property using the second pair of angles that we found. The only difference in using the
corresponding congruent angles 𝐴𝐸𝐷 and 𝐴𝐶𝐵 would be that the transversal would
instead be the line 𝐴𝐶. But this would still prove that
lines 𝐷𝐸 and 𝐵𝐶 are parallel.
