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Polygon Angle Sum - Problem 3
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
Recall that the sum of the measure of the angles in a polygon with n sides is (n - 2)*180°. If the angles of a given polygon are given in terms of x, it is possible to find the value of x by using this formula.
The number of sides that the polygon has is given, so first plug that value into the formula (n - 2)*180°. Add up the angles that are given in terms of variables. Then, solve for x using techniques from algebra.
An application of the polygon angle sum is a problem kind of like this, where you know how many sides, but you don’t know a variable. So let’s start by saying how many sides in this polygon do we have? Well it’s a pentagon, so we have 5 sides. I’d suggest that you have the polygon sum memorized all the way up to about a hexagon or a heptagon, but in case you forgot, you can always use your formula which says the number of sides minus 2 that quantity times 180 and where does 180 come from? It comes from the number of triangles that you can draw and the sum of the angles in a triangle is always 180 degrees.
So let’s substitute in 5 for n and make sure that we know what our angle sum should be. So we have 5 minus 2 which is 3, 3 times 180, 3 times 180 is 540, so if I add up all 5 of these angles, I should get 540 degrees. It sounds like an algebraic equation to me, so we’ll write 540 is equal to, just going in order 80 plus 140 plus 2x, plus the quantity of 2x plus 40, plus the quantity of x plus 5. So it looks like if we combine like terms and subtract something we can solve this pretty quickly.
So we’ll say 540 is equal to, I can combine 80 with 140 with 40 and with 5. Now if I add those all up I get 265. I can combine 2x plus 2x plus 1x, notice how I’m using different squiggles underneath to identify my like terms and that’s going to be 5x, 2 plus 2 plus 1, remember there is an invisible 1 there. So now we need to solve for x by subtracting 265 from both sides of our equation 540 minus 265 is 275, and last step here is to divide by 5, and 5 goes into 275 55 times. So find x, x is 55. Key thing here, knowing your angle sum and realizing that if you sum all these angles, it has to equal 540 degrees.
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