Hello, jmccoid!

What observations can you make about the sum of the angles of five pointed stars?

How can you prove your observation? Is this a **cyclic** pentagram? (not necessarily regular)

If so, it can be inscribed in a circle and skeeter's solution is excellent. Code:

A
* o *
* *
* *
E o o B
* *
* *
* *
* *
D o o C
* *
* * *

Draw the diagonals: .$\displaystyle AC,\, AD,\, BD,\, BE,\, CE$

An incribed angle is measured by one-half its intercepted arc.

. . $\displaystyle \begin{array}{ccc}\angle A &^m_= &\tfrac{1}{2}\overline{CD} \\ \\[-3mm]

\angle B & ^m_= & \tfrac{1}{2}\overline{DE} \\ \\[-3mm]

\angle C & ^m_= & \tfrac{1}{2}\overline{EA} \\ \\[-3mm]

\angle D & ^m_= & \tfrac{1}{2}\overline{AB} \\ \\[-3mm]

\angle E & ^m_= & \tfrac{1}{2}\overline{BC}

\end{array}$

Add: .$\displaystyle \angle A + \angle B + \angle C + \angle D + \angle E \;=\;\tfrac{1}{2}\overline{CD} + \tfrac{1}{2}\overline{DE} + \tfrac{1}{2}\overline{EA} + \tfrac{1}{2}\overline{AB} + \tfrac{1}{2}\overline{BC} $

$\displaystyle \text{Sum of the angles} \;=\;\tfrac{1}{2}\left(\overline{AB} + \overline{BC} + \overline{CD} + \overline{DE} + \overline{EA}\right) \;=\;\tfrac{1}{2}(360^o) \;=\;180^o$