## 3310.19 – A Pentangle

Here is a five-pointed star, also known as a pentangle. The five angles are all congruent. What is the sum of the five angles? (Does it matter whether they are congruent or not?)

## Solution

The measure of each angle is half the arc of circle opposite it. Adding the angles gives half the total angle of the circle, that is half of 360 degrees. The answer is 180 degrees.

The answer is the $same$ as long as the vertices of the pentangle lie on a circle but what if they don't? Hint 1: Any three vertices of the pentangle determine a circle. Pick the $smallest$ circle determined by three vertices. Then the two other vertices lie outside this circle. Analyze this situation. Hint 2: This is a perfect problem to study using GeoGebra or another geometry graphics program.