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Question

In the regular pentagon FGHIJ, extending the sides of the pentagon, as shown, forms a star. What is the measure of angle A in the figure?

Answer 1

Since this is a regular pentagon angle AFJ = angle AJF = 72° and therefore angle A = 180 - 144= 36°

Answer 2

First, we know that the interior angles of a pentagon add to (N-2)*180. For a pentagon, we have 5 sides, so the sum of the interior angles is (5-2)*180 degrees, or 540 degrees.

And since the pentagon is regular, all the interior angles are the same, so each interior angle is 540/5 = 108 degrees.

Consider the AJIH briangle. It is clearly an equilateral triangle, and the bottom angle is shared with the pentagon, and so is 108 degrees. The 2 smaller angles then must each be (180 - 108) / 2, or 36 degrees. Angle A is one of these two 36-degree angles.