Find 𝑥 and 𝑦.
Here we can see that angle 𝑦 is on the inside of the shape. And 𝑥 is on the outside of the shape. But exactly what shape is this? Since it has eight sides, this will be an octagon. So how would we go about finding an interior angle of this octagon and an exterior angle of this octagon?
Together, these angles make a straight line. So together, they add to 180 degrees. So if we can find one of them, we can subtract it from 180 and find the other. Since 𝑦 is on the inside of this octagon, we can use some things from the octagon to help us find it. So let’s begin by finding 𝑦.
A way that we can find an interior angle is using triangles. So how many triangles could fit inside this octagon? Here’s one, two, three, four, five, and six, so six total triangles. Now how many degrees are in a triangle? 180 degrees. And notice every angle of the triangles create all of the interior angles of the octagon. So all of the angles together make the octagon.
So we can take all six triangles and multiply by 180 degrees because that’s how many degrees are in a triangle. And then we get the total in degrees of the interior angles. So six times 180 is 1080. So that means all of these interior angles add to be a total of 1080.
So if we just want one of them, we need to divide by eight because there are eight angles in this octagon. So we get 135 degrees. So angle 𝑦 is equal to 135 degrees. And like we said before, angle 𝑥 and 𝑦 make a straight line just 180 degrees. So if we can plug in 135 for 𝑦, we can solve for 𝑥.
So let’s go ahead and subtract 135 from both sides of the equation. And we get that angle 𝑥 is equal to 45 degrees. So once again, 𝑥 is equal to 45 degrees and 𝑦 is equal to 135 degrees.