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The figure shown above consists of a shaded 9-sided...

The figure shown above consists of a shaded 9-sided polygon and 9 unshaded isosceles triangles…

Solution:

Since we have a 9-sided polygon (also known as a nonagon), we can first determine that the sum of the interior angles of the polygon is:

180(9 – 2) = 1260 degrees

Now, let’s let each of the base angles of the isosceles triangle with vertex angle labeled as “a” degrees be x degrees. Then the interior angles of the nonagon adjacent to these base angles of the isosceles triangle is 180 – x degrees each since each of these two interior angles is supplementary to the base angle of the isosceles triangle it is adjacent to. See diagram below:

gmat og shaded 9 sided polygon solution part one

Now we can argue that the base angle of x degrees is a vertical angle with the base angle of the “next” isosceles triangle. So that base angle is also x degrees. See diagram below:

gmat og shaded 9 sided polygon solution part two

Using the same argument we can say the base angles of all isosceles triangles must be x degrees and that all interior angles of the nonagon must be 180 – x degrees.

Thus, each interior angle of the nonagon is equal to 1260/9 = 140 degrees. Since 180 – x = 140, x = 40 degrees = base angle of each isosceles triangle. Thus, since a is the degree measure of the vertex angle of the isosceles triangle, a = 180 – 2(40) = 100 degrees.

Answer: A

About The Author

Jeffrey Miller is the head GMAT instructor for Target Test Prep. Jeff has more than fourteen years of experience in the business of helping students with low GMAT scores hurdle the seemingly impossible and achieve the scores they need to get into the top 20 business school programs in the world, including HBS, Stanford, Wharton, and Columbia. Jeff has cultivated many successful business school graduates through his GMAT instruction, and will be a pivotal resource for many more to follow.