The diagram shows four straight lines 𝐴𝐵, 𝐶𝐷, 𝐸𝐹, and 𝐺𝐻. 𝐴𝐵 and 𝐶𝐷 are parallel. 𝑅 is the point where the lines 𝐴𝐵, 𝐸𝐹, and 𝐺𝐻 intersect. The lengths of 𝑅𝐻 and 𝑅𝐹 are the same. Part a), calculate the size of angle 𝑥. You must show your working, which may be on the diagram.
It’s worth noting at this point that the diagram is not drawn accurately. So we cannot just measure the angles. So the first thing I do with this kind of question is mark on any information we’ve been given. So first of all, we’re told that 𝐴𝐵 and 𝐶𝐷 are parallel. So I’ve shown this using the orange arrowheads. So what we’re also told is that the lengths of 𝑅𝐻 and 𝑅𝐹 are the same. So I’ve marked these on. They’re shown here with these little orange lines. So this also tells us something else that’s interesting. We have an isosceles triangle. So therefore, with our isosceles triangle, we’re gonna have the two base angles being the same, which I’ve marked on here in orange.
So the first thing we can calculate is the fact that angle 𝐸𝑅𝐺 is gonna be equal to angle 𝐻𝑅𝐹. So they’re both gonna be equal to 80 degrees. And this is because they are vertically opposite angles. Remembering that we must give our reasoning for each stage of the problem. So we’ve said that they are both 80 degrees and given the reason why. And to help us understand what we mean by vertically opposite angles, well they mean the angles that are opposite each other when two lines meet. So we’re gonna see here that we’ve got a point. And the two angles that are orange are equal to each other. And the two angles that are green are equal to each other.
And when we say that they are vertically opposite, this doesn’t mean up and down as you might think with the word vertical. It means that they share a vertex. So they will share the corner point with each other. And that is the blue dot that I’ve put in there. Okay, great, so we now know what vertically opposite angles are. And we now know that we have our first angle. And next, we already know that angle 𝑅𝐹𝐻 and angle 𝑅𝐻𝐹 are equal because we already stated that these are gonna be the base angles of an isosceles triangle. And we can work these out because what we can do is 180 minus 80 divided by two. And that’s because the angles in a triangle sum to 180 degrees. So then we take away the 80 degrees that we had for the angle already found earlier, which was the angle 𝐻𝑅𝐹. And then, we divide by two because there are two angles. We’ve got 𝑅𝐹𝐻 and angle 𝑅𝐻𝐹.
So therefore, we can say that the angle 𝑅𝐹𝐻 and the angle 𝑅𝐻𝐹 are both gonna be equal to 50 degrees. And that’s because 180 minus 80 is 100. 100 over two is 50. And we’ve said already that the angles in a triangle sum to 180 degrees. And we’d also add in to our reasoning that the base angles of isosceles triangles are equal. Well, what we could do now is use this to work out angle 𝑥. Cause we can say that 180 minus angle 𝑅𝐻𝐹 is gonna be equal to 𝑥. And this is because the angles on a straight line sum to 180 degrees. And we can see that if we put angle 𝑥 and angle 𝑅𝐻𝐹 together, we have a straight line. Also, it’s worth reinforcing why this is the case. As you can see in pink, I’ve drawn a line that shows what’s happening between these two angles.
Well, we can see that the pink line I’ve drawn, in fact, makes a semicircle. And we can say that a straight line is a semicircle. So therefore, it’s gonna be half a circle. Well, we know that the angles in a circle add up to 360 degrees. So therefore, the angles in a semicircle must be 180 degrees. So thus, the angles on a straight line sum to 180 degrees. Well, we already know that angle 𝑅𝐻𝐹 is equal to angle 𝑅𝐹𝐻. And they’re both equal to 50 degrees. So then, 𝑥 is gonna be equal to 180 minus 50, which gives us a final answer to part a) of 130 degrees. And we found that by showing our working clearly and also giving reasoning for each step of the calculation. So now what we’re gonna do is move on to part b).
And in part b), we’re asked to calculate the size of angle 𝑦. Well, to help us calculate the size of angle 𝑦, we’re gonna have to use some more of our geometric properties. Well, first of all, we can say that angle 𝐴𝑅𝐻 is gonna be equal to angle 𝑅𝐻𝐹. And they’re both gonna be equal to 50 degrees. And this is because they are alternate angles. So alternate angles, that I’ve shown in the small sketch here, are angles that are in parallel lines, as we can see here. And they are also sometimes known as Z angles because they make a Z, as the ones in orange do. They can also make a backward Z, as we’ve shown here, with the green angles. So these would be the same, and so would the orange angles be the same. So that’s alternate angles. And that’s the reason why we know that these two angles are the same.
We could’ve also found angle 𝐴𝑅𝐻 using another property. And that is that if we’ve got 180 minus 130, this is gonna be equal to 50 degrees. And this is using the property called supplementary angles. And supplementary angles are the interior angles found between two parallel lines, as I’ve shown here. So we’ve got the green and the orange angle. And they sum to 180 degrees. Let’s think about why that might be. Well, if I straightened up the line that transverses our parallel lines, then what we’d have is something like we’d got in the second diagram.
But here, we could see that if that was a directly vertical line and it was perpendicular to the parallel lines, then what we’d have is two right angles. Well, two right angles must sum to 180 degrees. So therefore, if we push it over, what’s gonna happen is that the top angle is going to get bigger in the same proportion that the bottom angle is going to get smaller. But their sum will still remain at 180 degrees. And it’s worth noting that if we have used supplementary angles, the supplementary angle with 𝐴𝑅𝐻 would have been 𝑅𝐻𝐶.
Okay, so now what’s the next step? Well, now what we can do is find out the angle 𝑦. And we can do that because we can say the size of angle 𝑦 is gonna be equal to angle 𝐴𝑅𝐻, which is 50 degrees. And once more, we need to include our reasoning. Our reasoning again would be because they are vertically opposite angles. And we’ve already explained what those are. But we can see that that’s the case in our diagram. So therefore, because they’re equal, 𝑦 is gonna be equal to 50 degrees. And we’ve solved part a) because we’ve found that 𝑥 was equal to 130 degrees and part b) because we’ve found that 𝑦 was equal to 50 degrees.