Which of the following rays is
parallel to ray 𝐴𝐸? Option (A) ray 𝐵𝐺, option (B) ray
𝐶𝐼, option (C) ray 𝐵𝐹, option (D) ray 𝐶𝐻, or option (E) ray 𝐴𝐷.
Let’s start by identifying the ray
𝐴𝐸 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 𝐴𝐶, which contains the
point 𝐵. We could draw a very similar line
like this. Let’s call this the line 𝑃𝑄. We could then add the two parallel
rays, 𝑃𝑆 and 𝑄𝑇.
And what would that tell us about
these two angle measures at vertices 𝑃 and 𝑄? 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
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
So let’s return to the figure and
the ray 𝐴𝐸. We can label a point on the line
here with a letter such as 𝑇. Then, we can refer to the angle
between the ray 𝐴𝐸 and the line segment 𝐴𝑇 as the angle 𝑇𝐴𝐸. We can then calculate that the
measure of this angle 𝑇𝐴𝐸 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 𝐴𝐶 and another ray that also equal 58
degrees. We could work out that the measure
of angle 𝐴𝐵𝐺 is 38 degrees plus 21 degrees, which is 59 degrees. Notice that the measures of angles
𝑇𝐴𝐸 and 𝐴𝐵𝐺 are not equal, so the ray 𝐵𝐺 is not parallel to the ray
Next, we can calculate the measure
of angle 𝐵𝐶𝐼, which is equal to the sum of 19 degrees and 39 degrees. That’s 58 degrees.
Now we do have two congruent
angles, since the measure of angle 𝑇𝐴𝐸 is equal to the measure of angle
𝐵𝐶𝐼. Therefore, we can say that the ray
𝐶𝐼 is parallel to the ray 𝐴𝐸. This is the answer given in option
(B). But let’s check the other options
just to be sure there are no others.
We’ve already determined that ray
𝐵𝐺 is not parallel to ray 𝐴𝐸, so we can eliminate answer option (A).
Next, we can consider the ray 𝐵𝐹,
which means we can check the measure of angle 𝐴𝐵𝐹. But the measures of 58 degrees and
21 degrees are not equal, so the rays 𝐴𝐸 and 𝐵𝐹 are not parallel.
Similarly, to check if ray 𝐶𝐻 is
parallel to ray 𝐴𝐸, we can note that the measures of 58 degrees for angle 𝑇𝐴𝐸
is not equal to the measure of 39 degrees for angle 𝐵𝐶𝐻. Therefore, ray 𝐶𝐻 is not parallel
to ray 𝐴𝐸 either.
Finally in option (E), we are
comparing the rays 𝐴𝐷 and 𝐴𝐸. But both of these rays extend from
the same point 𝐴, 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 𝐴𝐸 is ray 𝐶𝐼.