In the following figure, what is the measure of angle 𝐹𝐴𝐵 in the hexagon 𝐴𝐵𝐶𝐷𝐸𝐹?
First, let’s identify the hexagon. 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, 𝐹, which then connects back to 𝐴. That’s our hexagon. Secondly, let’s identify the angle we want to know the measure of, which is angle 𝐹𝐴𝐵. We want to know how many degrees this angle would be. Angle 𝐹𝐴𝐵 is an interior angle for this hexagon. And before we can solve for our missing angle, we have to remember a formula for the sum of interior angles of polygons with 𝑛 sides.
We take the number of sides in our polygon and subtract two and then multiply that by 180 degrees. This will give us the sum of all the interior angles in our polygon. Our hexagon has six sides. So we’ll have six minus two times 180 degrees. Four times 180 degrees equals 720 degrees. And so, we can say that all of the angles, all six of the interior angles in our hexagon, must add up to 720.
The measure of angle 𝐹𝐴𝐵, we don’t know. But we do know the other five angles. We remember that the square means that this angle measures 90 degrees. Angle 𝐶𝐵𝐴 is 90 degrees. And so, we can say that the measure of angle 𝐹𝐴𝐵 plus 90 degrees plus 150 degrees plus 129 degrees plus 120 degrees plus 71 degrees must add up to 720 degrees. At this point, it’s probably worth it to do a check that we have all six angles accounted for. One is our missing angle. And then angles two through six, we know the measure of. So we’re ready to solve.
90 plus 150 plus 129 plus 120 plus 71 equals 560 degrees. And so, we can say that our missing angle plus 560 degrees has to equal 720 degrees. And so, we’ll subtract 560 degrees from 720 degrees. This will tell us that the measure of angle 𝐹𝐴𝐵 is 160 degrees. 720 degrees minus 560 degrees equals 160. We can add that to our diagram for a final answer of 160 degrees.