Video Transcript
In the given figure, if the measure
of angle 𝐴𝐷𝐶 equals the measure of angle 𝐴𝐸𝐵, then which of the following is
true? (A) 𝐷𝐶 equals 𝐴𝐸. (B) 𝐷𝐶 equals 𝐵𝐸. (C) 𝐷𝐶 equals 𝐸𝐶. (D) 𝐴𝐷 equals 𝐵𝐸. Or (E) 𝐷𝐵 equals 𝐴𝐸.
Let’s begin by noting that we are
given that the measures of angles 𝐴𝐷𝐶 and 𝐴𝐸𝐵 are equal. And these are colored on the
diagram. We also have a pair of congruent
sides marked on the diagram. So, we can write that line segments
𝐴𝐷 and 𝐴𝐸 are congruent.
Now let’s consider two triangles
within this larger triangle: triangle 𝐴𝐵𝐸 and triangle 𝐴𝐷𝐶. And sometimes, a separate sketch
can be useful if we find it difficult to see these smaller triangles within a larger
shape. In each triangle, we can still see
the pair of congruent sides and the given congruent angle measures. And we may also notice that the
angle at vertex 𝐴 appears in both triangles. We can write that the measure of
angle 𝐵𝐴𝐸 equals the measure of angle 𝐶𝐴𝐷.
Now, we can see that in each
triangle, we have two pairs of congruent angles and the pair of included sides
between them are congruent. Therefore, this proves that
triangles 𝐴𝐶𝐷 and 𝐴𝐵𝐸 are congruent, using the ASA or angle-side-angle
congruency criterion. We can note that the options that
we were given concern the congruency of sides. So let’s think about what sides we
can say are congruent by using the congruent triangles.
We already know that line segments
𝐴𝐷 and 𝐴𝐸 are congruent. And now we know that line segments
𝐷𝐶 and 𝐵𝐸 are congruent because these are corresponding. The third side in each triangle,
that’s line segments 𝐴𝐵 and 𝐴𝐶, are also congruent. The only one of these statements
that appears in the answer options is that 𝐷𝐶 equals 𝐵𝐸. But let’s see if any of the others
are also true.
In option (A), we can find line
segments 𝐷𝐶 and 𝐴𝐸 here on the figure. And we can observe that we have no
information to prove that these pair of line segments are congruent. The same is true for option
(C). We cannot prove that line segments
𝐷𝐶 and 𝐸𝐶 are congruent. Line segments 𝐴𝐷 and 𝐵𝐸 also
cannot be proved to be congruent. And neither can the line segments
𝐷𝐵 and 𝐴𝐸. Therefore, the only statement which
is true is that 𝐷𝐶 equals 𝐵𝐸.
