So in this question, we’ve been given a diagram and we’re asked to calculate the value of 𝑥, which is the size of one of the angles in the diagram. We can see as well that we have a pair of parallel lines as indicated by the blue arrows along their lengths. So within this question, we’re going to be using angle rules about angles in parallel lines.
Now the first thing I’m going to do is actually add another line to the diagram, a third line that is parallel to the two existing lines. So I’ve added in this line here in orange, and now I have two pairs of parallel lines: I have one in the top half of the diagram and one in the lower half of the diagram. And in each case, I have one angle that has been marked for me.
So my strategy for answering this question is going to be this: I’d like to calculate this angle here first of all, the angle I have marked in green which I’m going to refer to as 𝑤. I’d then like to calculate this angle here, the angle I’ve marked in pink which I’m going to refer to as 𝑦. Finally, I’m going to use the fact that angles 𝑥, 𝑦, and 𝑤 are all situated around a point in order to calculate angle 𝑥.
So those are the three stages of the method that I’m going to follow. Let’s start off then by calculating angle 𝑤. If I look then at the top half of the diagram, I’ll see the angle 𝑤 and the angle marked as 67 degrees of a particular type of angle in parallel lines. They both sit inside the parallel lines and to the right of the transversal, the line that cuts the parallel lines.
This means that these two angles are cointerior angles. Now there’s a key fact about cointerior angles that will help me here, and it’s that cointerior angles are supplementary to one another, which means that the sum of those two angles is 180 degrees. So I can calculate angle 𝑤 by subtracting the angle of 67 degrees from 180. So we have the angle 𝑤 is 113 degrees.
So there’s step one of the method completed. Now I want to calculate angle 𝑦. If I look at the bottom half the diagram, I can see that I have the same situation again. This time angle 𝑦 and the angle of 65 degrees are cointerior angles. Therefore these two angles are supplementary, which means again they must sum to 180 degrees. So angle 𝑦 can be calculated by subtracting 65 degrees from 180 degrees, and therefore we have the angle 𝑦 is equal to 115 degrees.
Finally, I want to calculate 𝑥. And as I mentioned earlier, I’m going to use the fact that angles 𝑥, 𝑦, and 𝑤 are all situated around a point, which means that they must sum to 360 degrees. So I can calculate angle 𝑥 by subtracting 𝑤 and 𝑦 from 360. We have that 𝑥 is equal to 360 degrees minus 113 degrees minus 115 degrees, and therefore 𝑥 is equal to 132 degrees.
So adding the extra line in the diagram created two pairs of parallel lines that we could then use in order to solve this problem. The key fact that we use was the cointerior angles in parallel lines are supplementary, so their sum is 180 degrees. We also use the fact that angles around a point sum to 360 degrees.