### Video Transcript

The diagram shows four straight lines ๐ด๐ต, ๐ถ๐ท, ๐ธ๐น, and ๐บ๐ป. ๐ด๐ต and ๐ถ๐ท are parallel. ๐
is the point where the lines ๐ด๐ต, ๐ธ๐น, and ๐บ๐ป intersect. The lengths of ๐
๐ป and ๐
๐น are the same. Part a), calculate the size of angle ๐ฅ. You must show your working, which may be on the diagram.

Itโs 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 weโve been given. So first of all, weโre told that ๐ด๐ต and ๐ถ๐ท are parallel. So Iโve shown this using the orange arrowheads. So what weโre also told is that the lengths of ๐
๐ป and ๐
๐น are the same. So Iโve marked these on. Theyโre shown here with these little orange lines. So this also tells us something else thatโs interesting. We have an isosceles triangle. So therefore, with our isosceles triangle, weโre gonna have the two base angles being the same, which Iโve marked on here in orange.

So the first thing we can calculate is the fact that angle ๐ธ๐
๐บ is gonna be equal to angle ๐ป๐
๐น. So theyโre 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 weโve 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 weโre gonna see here that weโve 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 doesnโt 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 Iโve 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 ๐
๐น๐ป and angle ๐
๐ป๐น 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 thatโs 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 ๐ป๐
๐น. And then, we divide by two because there are two angles. Weโve got ๐
๐น๐ป and angle ๐
๐ป๐น.

So therefore, we can say that the angle ๐
๐น๐ป and the angle ๐
๐ป๐น are both gonna be equal to 50 degrees. And thatโs because 180 minus 80 is 100. 100 over two is 50. And weโve said already that the angles in a triangle sum to 180 degrees. And weโd 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 ๐ฅ. Cause we can say that 180 minus angle ๐
๐ป๐น is gonna be equal to ๐ฅ. And this is because the angles on a straight line sum to 180 degrees. And we can see that if we put angle ๐ฅ and angle ๐
๐ป๐น together, we have a straight line. Also, itโs worth reinforcing why this is the case. As you can see in pink, Iโve drawn a line that shows whatโs happening between these two angles.

Well, we can see that the pink line Iโve drawn, in fact, makes a semicircle. And we can say that a straight line is a semicircle. So therefore, itโs 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 ๐
๐ป๐น is equal to angle ๐
๐น๐ป. And theyโre both equal to 50 degrees. So then, ๐ฅ 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 weโre gonna do is move on to part b).

And in part b), weโre asked to calculate the size of angle ๐ฆ. Well, to help us calculate the size of angle ๐ฆ, weโre gonna have to use some more of our geometric properties. Well, first of all, we can say that angle ๐ด๐
๐ป is gonna be equal to angle ๐
๐ป๐น. And theyโre both gonna be equal to 50 degrees. And this is because they are alternate angles. So alternate angles, that Iโve 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 weโve shown here, with the green angles. So these would be the same, and so would the orange angles be the same. So thatโs alternate angles. And thatโs the reason why we know that these two angles are the same.

We couldโve also found angle ๐ด๐
๐ป using another property. And that is that if weโve 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 Iโve shown here. So weโve got the green and the orange angle. And they sum to 180 degrees. Letโs think about why that might be. Well, if I straightened up the line that transverses our parallel lines, then what weโd have is something like weโd 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 weโd have is two right angles. Well, two right angles must sum to 180 degrees. So therefore, if we push it over, whatโs 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 itโs worth noting that if we have used supplementary angles, the supplementary angle with ๐ด๐
๐ป would have been ๐
๐ป๐ถ.

Okay, so now whatโs the next step? Well, now what we can do is find out the angle ๐ฆ. And we can do that because we can say the size of angle ๐ฆ is gonna be equal to angle ๐ด๐
๐ป, 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 weโve already explained what those are. But we can see that thatโs the case in our diagram. So therefore, because theyโre equal, ๐ฆ is gonna be equal to 50 degrees. And weโve solved part a) because weโve found that ๐ฅ was equal to 130 degrees and part b) because weโve found that ๐ฆ was equal to 50 degrees.