Are the lines 𝐿 one and 𝐿 two parallel?
So as you can see in the diagram here, we have two lines. And we need to see: Are they parallel? We’ve also got some angles given to us. Now we’re gonna use these angles to help us prove whether the lines 𝐿 one and 𝐿 two are parallel or not.
To help us understand how we’re going to do that, we’re first gonna look at some properties of parallel lines. So, first of all, we have something called alternate angles. And these are sometimes known as Z angles, cause it looks like they can make a Z shape. So I’ve marked them here in pink. These two angles will be equal because, as I said, they’re alternate angles.
Another relationship we know in parallel lines is if we have some angles here that I’ve marked in red, they look like F angles, as they’re sometimes known. These are called corresponding angles. And these are also equal. So we can see that we’ve actually, so far, got two pairs of equal angles. And the other relationship we’re going to use is that if we have two angles that are inside, as we have here, they’re known as supplementary or complementary angles. And these two sum to 180 degrees. Remembering that each of these relationships only work if we are dealing with parallel lines. So I’ve shown that clearly with these pink arrows.
And to help us understand why the supplementary angles work, let’s consider a rectangle or a square. So here, I’ve drawn a rectangle. We know that we got parallel sides. And we know that each of the angles is 90 degrees. So therefore, we know that the sum of this pair of angles would be 180 degrees, cause 90 add 90 makes 180 degrees.
Now if we think about what would happen if we moved the parallel sides but kept them parallel, we could see that the individual angles would change. However, their sum would not. Because in the same proportion that one angle got smaller, so it got less than 90 degrees, in the same proportion, the other angle will get greater than 90 degrees. So therefore, they’d always sum to 180. Okay, great. So we now know our relationships for parallel lines. Let’s use them to see if we can work out whether the lines 𝐿 one and 𝐿 two are parallel.
So the first thing that I’m going to do is mark on any angles I can, using information that we know. So first of all, I’ve got this angle here 𝑥. Well, 𝑥 is an angle in a triangle. And we see that the triangle is made up of 𝑥, a right angle, and 35 degrees. So therefore, we can say that the value of 𝑥 is gonna be equal to 180 minus 90 minus 35. And this is gonna give us a value of 55 degrees. And we know this because the angles in a triangle sum to 180 degrees. Remember, in a question like this, you must give your reasons. So I’ve given my reasoning here: it’s because the angles in a triangle sum to 180 degrees.
So next, we want to find out angle 𝑦. And we can find angle 𝑦, because angle 𝑦 is gonna be equal to 180 minus 90 minus 55. And this is gonna be equal to 35 degrees. And we know this because the angles on a straight line sum to 180 degrees. And we know that they sum to 180 degrees because it’s half a circle.
Okay, so now, are there any other values we can add on? Well, finally, we can mark on angle 𝑧. We can mark on angle 𝑧 because, again, we’re gonna use angles on a straight line sum up to 180 degrees. So 𝑧 is gonna be equal to 180 minus 55, which gives us 125 degrees. So now, we’ve marked on all the angles we can. Now let’s see if the lines 𝐿 one and 𝐿 two are parallel. Well, first of all, we can check out the alternate angles; are these equal?
Well, yes, we can see they are, cause we’ve got 55 degrees and 55 degrees. Then we’ve got 35 degrees and 35 degrees. So, that shows that they are parallel. And also, we have a pair of supplementary or complementary angles that sum to 180 degrees. That’s cause we’ve got 55 and 125. Add these together we get 180, so 180 degrees. So therefore, 𝐿 one and 𝐿 two must be parallel. And we’ve shown the reasons why.