Brian McCall

**Univ. of Wisconsin**

J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

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Given two adjacent polygons, it is possible to find the measure of an angle of a regular polygon and an adjacent triangle sharing a side. Recall the sum of the measures of an angle in a regular polygon is (n - 2)*180°/n.

If told that an angle in a polygon is part of a linear pair with the base angle of an isosceles triangle, you know that the measure of the base angle is 180° minus the measure of the angle of the polygon. Recall that the sum of the angles in an isosceles triangle is 180°, and the base angles are congruent. So, the measure of the third angle in an isosceles triangle is 180° - 2*(base angle measure).

A more difficult problem would involve something we aren’t given any numbers. All you’re going off of here is knowing that you have a pentagon. So let’s find the measure of one of those angles.

To do that we’re going to do the quantity of n minus 2 times 180 all divided by n, and since we have five sides, I’m going to substitute in five for n, so we have 5 minus 2 times 180 all divided by 5. Now it might be a good idea to memorize up to a hexagon of what a regular polygon would have.

So in this case you’re going to find out that you get 108 degrees, so y is 108 degrees and I’m going to come over here and I’m going to write that y is 108. How are we going to find x though?

Well we see that we have a linear pair here and we have another linear pair which means that these two must be congruent, so we have an isosceles triangle and if this angle right here is 108 degrees and these two are a linear pair, that means that this must be 72 degrees. So if I’m looking at this triangle, I see that the sum is 180 degrees and I’m adding 72 plus 72 plus x, why did I add 72 twice? Because our base angles are congruent, so we get 180 equals 144 plus x and to solve, we’re going to subtract 144 and you see that x must be 36.

So I’m going to come over here and write that x is 36 degrees. The key to this problem is remembering that in an equiangular polygon, our formula is quantity of n minus 2 times 180 divide by n. Once you knew that we use linear pairs and triangle angle sum to find x.