Question Video: Identifying a Pair of Parallel Rays Mathematics • 8th Grade

Which of the following rays is parallel to ray 𝐴𝐸? [A] Ray 𝐡𝐺 [B] Ray 𝐢𝐼 [C] Ray 𝐡𝐹 [D] Ray 𝐢𝐻 [E] Ray 𝐴𝐷


Video Transcript

Which of the following rays is parallel to ray 𝐴𝐸? Option (A) ray 𝐡𝐺, option (B) ray 𝐢𝐼, option (C) ray 𝐡𝐹, option (D) ray 𝐢𝐻, 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 𝐴𝐢, which contains the point 𝐡. 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 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 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 𝐴𝐢 and another ray that also equal 58 degrees. We could work out that the measure of angle 𝐴𝐡𝐺 is 38 degrees plus 21 degrees, which is 59 degrees. Notice that the measures of angles 𝑇𝐴𝐸 and 𝐴𝐡𝐺 are not equal, so the ray 𝐡𝐺 is not parallel to the ray 𝐴𝐸.

Next, we can calculate the measure of angle 𝐡𝐢𝐼, 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 𝐡𝐢𝐼. Therefore, we can say that the ray 𝐢𝐼 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 𝐡𝐺 is not parallel to ray 𝐴𝐸, so we can eliminate answer option (A).

Next, we can consider the ray 𝐡𝐹, which means we can check the measure of angle 𝐴𝐡𝐹. But the measures of 58 degrees and 21 degrees are not equal, so the rays 𝐴𝐸 and 𝐡𝐹 are not parallel.

Similarly, to check if ray 𝐢𝐻 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 𝐡𝐢𝐻. Therefore, ray 𝐢𝐻 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 𝐢𝐼.

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