In the following figure, the lines 𝑘 and 𝑙 are parallel and the lines 𝑎 and 𝑏 intersect them such that the point 𝐴 lies on the lines 𝑘, 𝑎, and 𝑏. Given that the measure of angle one is equal to 32 degrees and the measure of angle two is equal to 72 degrees, what is the measure of angle three?
So with any question like this, the first thing I do is mark on any of the information I’ve been given. So first of all, we’re given that the measure of angle one is equal to 32 degrees. So I’ve added that to the diagram. Then, we’re also told that the measure of angle two is equal to 72 degrees. So I’ve also added that to the diagram. Now what we need to do is mark on any values that we can find using the properties of our parallel lines.
So when we’re dealing with parallel lines, we have some relationships that we know. The first of these is alternate angles. And what I mean by this is the angles that are both orange will be equal to each other and the angles that are pink will be equal to each other. Alternate angles are also sometimes known as Z angles and that’s because they make a Z with each other: either a forward Z as I’ve drawn here in orange or if we’re looking at the pink angles, it would be a backward Z.
Now another relationship that we have with parallel lines is that corresponding angles are the same. And these are corresponding angles. So we’ve got the two orange angles would be the same and the two pink angles would be the same. And that’s because if we cut the line in half and move the angle upwards, it would be corresponding. And that’s because as we’ve said both of those lines are parallel. Corresponding angles are also sometimes known as F angles. And that’s because as I’ve shown here, if we have an F shape where we’ve got the orange angles, the ones that would be corresponding would fit into this shape.
And the final relationship we have is that supplementary angles sum to 180 degrees. And what this means is that the angles on the insides of our parallel lines when added together will equal 180 degrees. To illustrate what this might be, I’ve drawn a square. So if we take a look at this square, we can see that all of the angles are 90 degrees. So in that case, if I added these two angles together, I get 180 degrees because it’d be 90 plus 90, which is 180. Also in a square, we know that we have two pairs of parallel sides.
Okay, so how does this help us to show that supplementary angles sum to 180 degrees? So if we pushed our square over, so as we can see here it’d be on a slant, then the top and bottom angles have increased and decreased in the same ratio. So therefore, when added together, they still must be equal to 180 degrees. However, one of them is going to be smaller and one of them is going to be bigger than 90 degrees. Okay, so let’s use these to add some more angles to our diagram.
Then, the next angle that we can work out is angle 𝑥 I’ve marked on here. And that’s because angle 𝑥 plus 72 must be equal to 180. So therefore, the measure of angle 𝑥 is gonna be 180 minus 72. And that’s because they’re angles on a straight line. So therefore, the measure of angle 𝑥 is gonna be equal to 180 degrees. So I’ve added that onto the diagram as well.
We could have also worked that out using the fact that the two angles I’ve marked here in pink are supplementary angles. So they must sum to 180 degrees. So then next, I’ve marked on another angle and this is an angle that’s 72 degrees. And that’s because this is a corresponding angle. We can see that it’s corresponding to the original 72 degrees that we were given.
It is only in fact this last angle that we need to find the measure of angle three. Because now we’ve marked this on, we can see that 32 degrees, this 72 degrees, and angle three all make a straight line. So therefore, they must sum to 180 degrees. So therefore, we can say that the measure of angle three is gonna be equal to 180 minus 72 minus 32. And again, that’s because it’s angles on a straight line. So therefore, we can see that the measure of angle three is gonna be equal to 76 degrees.
And as you can see, at each stage, I’ve shown the reasoning why. And whenever you’re solving a problem like that, you should do that — show your reasoning. So as we’ve said, we can say that the measure of angle three is 76 degrees.