### Video Transcript

Which of the following rays is
parallel to ray π΄πΈ? Option (A) ray π΅πΊ, option (B) ray
πΆπΌ, option (C) ray π΅πΉ, option (D) ray πΆπ», or option (E) ray π΄π·.

Letβs start by identifying the ray
π΄πΈ in the figure, which is marked here in orange. Now, one of the important features
in this diagram is the fact that we have this straight line π΄πΆ, which contains the
point π΅. We could draw a very similar line
like this. Letβs call this the line ππ. We could then add the two parallel
rays, ππ and ππ.

And what would that tell us about
these two angle measures at vertices π and π? Well, we can recall that in a
transversal of parallel lines, corresponding angles are congruent. And really important for this
question is knowing that the converse of this statement is also true. If we want to determine if two
given lines are parallel, then we can check if the corresponding angles are
congruent. If they are, then the lines are
parallel.

We can write this in the following
way. If corresponding angles formed by a
transversal cutting two lines are congruent, then the lines cut by the transversal
are parallel.

So letβs return to the figure and
the ray π΄πΈ. We can label a point on the line
here with a letter such as π. Then, we can refer to the angle
between the ray π΄πΈ and the line segment π΄π as the angle ππ΄πΈ. We can then calculate that the
measure of this angle ππ΄πΈ is the sum of the two angle measures within it, which
is 26 degrees plus 32 degrees, a total of 58 degrees.

So now we can determine if there
are any other angles made with the line π΄πΆ and another ray that also equal 58
degrees. We could work out that the measure
of angle π΄π΅πΊ is 38 degrees plus 21 degrees, which is 59 degrees. Notice that the measures of angles
ππ΄πΈ and π΄π΅πΊ are not equal, so the ray π΅πΊ is not parallel to the ray
π΄πΈ.

Next, we can calculate the measure
of angle π΅πΆπΌ, which is equal to the sum of 19 degrees and 39 degrees. Thatβs 58 degrees.

Now we do have two congruent
angles, since the measure of angle ππ΄πΈ is equal to the measure of angle
π΅πΆπΌ. Therefore, we can say that the ray
πΆπΌ is parallel to the ray π΄πΈ. This is the answer given in option
(B). But letβs check the other options
just to be sure there are no others.

Weβve already determined that ray
π΅πΊ is not parallel to ray π΄πΈ, so we can eliminate answer option (A).

Next, we can consider the ray π΅πΉ,
which means we can check the measure of angle π΄π΅πΉ. But the measures of 58 degrees and
21 degrees are not equal, so the rays π΄πΈ and π΅πΉ are not parallel.

Similarly, to check if ray πΆπ» is
parallel to ray π΄πΈ, we can note that the measures of 58 degrees for angle ππ΄πΈ
is not equal to the measure of 39 degrees for angle π΅πΆπ». Therefore, ray πΆπ» is not parallel
to ray π΄πΈ either.

Finally in option (E), we are
comparing the rays π΄π· and π΄πΈ. But both of these rays extend from
the same point π΄, so we know that they cannot be parallel. If two parallel lines share one
point, then they are coincident and share all points. And we can see from the diagram
that this is not the case.

Therefore, the only ray in the
given diagram which is parallel to ray π΄πΈ is ray πΆπΌ.