This is a regular polygon. Find the measure of angle 𝑥.
If this is a regular polygon, then it is equiangular and equilateral. Equiangular means all the angles are equal in measure. And equilateral means all side lengths are equal in measure. This means angle 𝑥 will be equal to this angle, this angle, this angle, and this angle. Now, there is a formula to use to find one angle of a regular polygon. It’s 𝑛 minus two times 180 all divided by 𝑛, where 𝑛 is the number of sides.
So to find the number of sides, we simply need to count them: one, two, three, four, and five. This means we’re working with a pentagon. So this means we need to plug in five for 𝑛. So we have five minus two times 180 all divided by five. So five minus two is three. And three times 180 is 540. And 540 divided by five is 108. Therefore, 𝑥 is equal to 108 degrees.
Now let’s say that we didn’t remember the formula. But we did remember that in a triangle, there are 180 degrees. And if we would take our shape and split it into triangles, we will know how many degrees total the shape would have. So we need to pick a vertex. How about this one? And then from this vertex go to other corners, as many as we can, and make as many triangles as possible inside the shape: one, two, and three. So there are three triangles. And notice, all of the angles of the triangles — for example, triangle one — are all on the vertices of this polygon, same for the second triangle and the third triangle. There’s really nothing in the center.
So there are three triangles at 180 degrees a piece. So we take 180 and multiply by three. So we get 540 degrees. So this represents all of the angles added together. So if we only want one of them and they’re all equal, we can divide by the number of angles that there are. And there are: one, two, three, four, five. So we divide by five and find that each angle is equal to 108 degrees, just as we had before.
So once again, the measure of angle 𝑥 is equal to 108 degrees.