### Video Transcript

π΄π΅πΆπ·πΈπΉ is a regular hexagon. πΈπ is a straight line. Show that triangle πΆππ· is a right-angled triangle.

So weβre looking at triangle πΆππ·, and weβve been asked to show that this is a right-angled triangle, which means we need to consider the size of its angles. We know that one angle is 30 degrees, and we need to consider the other two. However, we havenβt got any other information about the angles in this triangle. So we need to look at the hexagon first.

This hexagon has six interior angles, and we noticed that one of them, angle πΈπ·πΆ, is on a straight line with one of the angles in our triangle. And this is because weβre told that πΈπ is a straight line. So if we can work out the size of angle πΈπ·πΆ, this will also tell us something about angle πΆπ·π, which is in our triangle.

Now we actually have a formula for calculating the sum of the interior angles in a polygon. That just means a straight-sided shape with π sides. Itβs equal to 180 multiplied by π minus two. The reason for this formula comes from the fact that you can divide a polygon up into triangles.

So if you start at one corner or vertex and draw straight lines to all of the other corners or vertices which it isnβt already connected to, you form a certain number of triangles. In fact, you always form two less triangles than the number of sides that the polygon has. So the polygon Iβve drawn here has five sides, and we formed three triangles. This is where the π minus two part comes in, as we subtract two from the number of sides.

The interior angles of these triangles also make up the interior angles of the overall polygon. And as we know that the sum of angles in a triangle is 180 degrees, each triangle contributes 180 degrees to the total. So we have three multiplied by 180 degrees in this case or π minus two multiplied by 180 in the general case.

Now our shape π΄π΅πΆπ·πΈπΉ is a hexagon which has six sides. So this means that the value of π in this formula is going to be six. So we can substitute π equals six into our formula. And it tells us that the sum of the interior angles in the hexagon π΄π΅πΆπ·πΈπΉ is equal to 180 multiplied by six minus two degrees. So this is equal to 180 multiplied by four degrees, which is equal to 720 degrees.

Now we do have a calculator in this question to help with evaluating this. But if we didnβt, the easiest way to multiply a number by four is just to double it and double it again. So weβve worked out that the sum of the interior angles in π΄π΅πΆπ·πΈπΉ is 720 degrees.

But we wanted to just work out the size of one interior angle, specifically the angle πΈπ·πΆ. The key fact here is that π΄π΅πΆπ·πΈπΉ is a regular hexagon, which means that all of its sides are the same length. But more importantly, for this question, all of its interior angles are equal.

Now there are six interior angles. So to find the size of each of them, we need to divide the sum by six. So we have 720 degrees divided by six, which is 120 degrees. So this is the size of each of the interior angles of the hexagon, and itβs therefore the size of angle πΈπ·πΆ.

Now that we found the size of angle πΈπ·πΆ, we can work out the size of angle πΆπ·π because, remember, these angles are on a straight line. We know that angles on a straight line sum to 180 degrees. So we can subtract 120 degrees from 180 degrees, and it gives 60 degrees.

So great! Now we know two of the angles in our triangle πΆππ·. But remember, we wanted to show that it was a right-angled triangle. And neither of the angles that we know so far are right angles. So we need to calculate the size of the third angle in our triangle.

To find the third angle, we need to remember that angles in a triangle sum to 180 degrees. So we subtract the two known angles of 60 degrees and 30 degrees from 180 degrees, and it gives that angle π·πΆπ is equal to 90 degrees. So we can conclude that because angle π·πΆπ is 90 degrees, a right angle, this means that triangle πΆππ· is a right-angled triangle. Now all of the working out and all of the reasoning that weβve written on the screen is an essential part of our answer to this question.