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,
Therefore, by the Angle Addition Postulate,
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 .