Question Video: Identifying a Pair of Parallel Rays | Nagwa Question Video: Identifying a Pair of Parallel Rays | Nagwa

# Question Video: Identifying a Pair of Parallel Rays Mathematics • First Year of Preparatory School

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Which of the following rays is parallel to ray π΄πΈ? [A] Ray π΅πΊ [B] Ray πΆπΌ [C] Ray π΅πΉ [D] Ray πΆπ» [E] Ray π΄π·

04:32

### Video Transcript

Which of the following rays is parallel to ray π΄πΈ? Option (A) ray π΅πΊ, option (B) ray πΆπΌ, option (C) ray π΅πΉ, option (D) ray πΆπ», or option (E) ray π΄π·.

Letβs start by identifying the ray π΄πΈ 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΄πΆ, which contains the point π΅. We could draw a very similar line like this. Letβs call this the line ππ. We could then add the two parallel rays, ππ and ππ.

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 letβs return to the figure and the ray π΄πΈ. 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΄πΈ and the line segment π΄π as the angle ππ΄πΈ. We can then calculate that the measure of this angle ππ΄πΈ 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΄πΆ and another ray that also equal 58 degrees. We could work out that the measure of angle π΄π΅πΊ is 38 degrees plus 21 degrees, which is 59 degrees. Notice that the measures of angles ππ΄πΈ and π΄π΅πΊ are not equal, so the ray π΅πΊ is not parallel to the ray π΄πΈ.

Next, we can calculate the measure of angle π΅πΆπΌ, which is equal to the sum of 19 degrees and 39 degrees. Thatβs 58 degrees.

Now we do have two congruent angles, since the measure of angle ππ΄πΈ is equal to the measure of angle π΅πΆπΌ. Therefore, we can say that the ray πΆπΌ is parallel to the ray π΄πΈ. This is the answer given in option (B). But letβs check the other options just to be sure there are no others.

Weβve already determined that ray π΅πΊ is not parallel to ray π΄πΈ, so we can eliminate answer option (A).

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

Similarly, to check if ray πΆπ» is parallel to ray π΄πΈ, we can note that the measures of 58 degrees for angle ππ΄πΈ is not equal to the measure of 39 degrees for angle π΅πΆπ». Therefore, ray πΆπ» is not parallel to ray π΄πΈ either.

Finally in option (E), we are comparing the rays π΄π· and π΄πΈ. 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΄πΈ is ray πΆπΌ.

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