𝐴𝐵𝐶𝐷𝐸𝐹 is a regular hexagon. 𝐸𝑍 is a straight line. Show that triangle 𝐶𝑍𝐷 is a right-angled triangle.
So we’re looking at triangle 𝐶𝑍𝐷, and we’ve 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 haven’t 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 𝐸𝐷𝐶, is on a straight line with one of the angles in our triangle. And this is because we’re told that 𝐸𝑍 is a straight line. So if we can work out the size of angle 𝐸𝐷𝐶, this will also tell us something about angle 𝐶𝐷𝑍, 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 𝑛 sides. It’s equal to 180 multiplied by 𝑛 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 isn’t 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 I’ve drawn here has five sides, and we formed three triangles. This is where the 𝑛 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 𝑛 minus two multiplied by 180 in the general case.
Now our shape 𝐴𝐵𝐶𝐷𝐸𝐹 is a hexagon which has six sides. So this means that the value of 𝑛 in this formula is going to be six. So we can substitute 𝑛 equals six into our formula. And it tells us that the sum of the interior angles in the hexagon 𝐴𝐵𝐶𝐷𝐸𝐹 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 didn’t, the easiest way to multiply a number by four is just to double it and double it again. So we’ve worked out that the sum of the interior angles in 𝐴𝐵𝐶𝐷𝐸𝐹 is 720 degrees.
But we wanted to just work out the size of one interior angle, specifically the angle 𝐸𝐷𝐶. The key fact here is that 𝐴𝐵𝐶𝐷𝐸𝐹 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 it’s therefore the size of angle 𝐸𝐷𝐶.
Now that we found the size of angle 𝐸𝐷𝐶, we can work out the size of angle 𝐶𝐷𝑍 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 𝐶𝑍𝐷. 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 𝐷𝐶𝑍 is equal to 90 degrees. So we can conclude that because angle 𝐷𝐶𝑍 is 90 degrees, a right angle, this means that triangle 𝐶𝑍𝐷 is a right-angled triangle. Now all of the working out and all of the reasoning that we’ve written on the screen is an essential part of our answer to this question.