We have shown that the **angle sum of a star polygon** or a pentagram is . In this post, we discuss another proof of the star polygon angle sum theorem.

Let be a star polygon with angle measures and . Recall from the **Remote Exterior Angle Theorem** that the measure of the exterior angle of a triangle is the sum of the measures of its two remote interior angles.

We use this theorem to show that the angle sum of a star polygon is . In equation form, we want to show that

.

We start the proof by drawing ray . We let as shown in the next figure.

By the Remote Exterior Angle Theorem,

and .

Therefore, by the Angle Addition Postulate,

.

Since ,

.

Also, and are vertical angles, so their measures are equal. Therefore,

.

Now, if we add the interior angles of triangle , its angle sum is . Therefore, we have

.

But this is the sum of the interior angles of the star polygon. Therefore, the angle sum of a star polygon is equal to .