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Exterior Angles of a Polygon - Problem 1
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
An exterior angle is the angle constructed by extending a side of a polygon. The sum of the measures of the exterior angles of a polygon is always 360°. In an equiangular polygon (in which all angles are congruent), each equiangular angle has measure 360° divided by the number of sides.
So, if the exterior angles are given by expressions with variables, solve for the variable by adding up the expressions and setting the sum equal to 360°.
We can use what we know about exterior angles and their sum to find a sum of missing angles. So let’s go back and examine what we know about the sum of exterior angles. We found that no matter how many sides there are in a polygon, its sum will always be 360 degrees.
So getting back to this problem, we don’t have a regular triangle, also known as an equilateral triangle, but we do know that the three exterior angles must sum up to 360 degrees. So we’re going to write that equation that 360 is equal to the sum of the exterior angles. So I’m going to say it’s equal to x, which is that angle, plus x plus 1, plus your third exterior angle which is x plus 2. So now we just combine like terms and solve. So 360 is equal to 3x's plus 1 plus 2 which is 3.
Next step is to subtract 3, subtract 3 we get 357 is equal to 3x and last we’re going to divide by 3 and x is equal to 119. So over here solve for x, x is 119 degrees. But we also have to find a, b and c. So I’m going to erase x here and I’m going to write 119 degrees. I know that a and 119 must be supplementary which means a is 61 degrees. I come over here, I have x plus 1. If x is 119, that makes this angle, 120 because it is 1 more than 119 degrees, which means b must be 60 degrees. So our last one is x plus 2. If x is 119, 2 more than 119 is 121, which means c must be 59 degrees.
So the key to solving this problem first was remembering that the sum of the exterior angles of any polygon is 360 degrees. The second key was remembering that that we have linear pair of angles.
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