### Video Transcript

In the given figure, if the measure
of angle π΄π·πΆ equals the measure of angle π΄πΈπ΅, then which of the following is
true? (A) π·πΆ equals π΄πΈ. (B) π·πΆ equals π΅πΈ. (C) π·πΆ equals πΈπΆ. (D) π΄π· equals π΅πΈ. Or (E) π·π΅ equals π΄πΈ.

Letβs begin by noting that we are
given that the measures of angles π΄π·πΆ and π΄πΈπ΅ 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
π΄π· and π΄πΈ are congruent.

Now letβs consider two triangles
within this larger triangle: triangle π΄π΅πΈ and triangle π΄π·πΆ. 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 π΄ appears in both triangles. We can write that the measure of
angle π΅π΄πΈ equals the measure of angle πΆπ΄π·.

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 π΄πΆπ· and π΄π΅πΈ 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 letβs think about what sides we
can say are congruent by using the congruent triangles.

We already know that line segments
π΄π· and π΄πΈ are congruent. And now we know that line segments
π·πΆ and π΅πΈ are congruent because these are corresponding. The third side in each triangle,
thatβs line segments π΄π΅ and π΄πΆ, are also congruent. The only one of these statements
that appears in the answer options is that π·πΆ equals π΅πΈ. But letβs see if any of the others
are also true.

In option (A), we can find line
segments π·πΆ and π΄πΈ 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
π·πΆ and πΈπΆ are congruent. Line segments π΄π· and π΅πΈ also
cannot be proved to be congruent. And neither can the line segments
π·π΅ and π΄πΈ. Therefore, the only statement which
is true is that π·πΆ equals π΅πΈ.