# Thread: 5 pointed star

1. ## 5 pointed star

what observations can you make about the sum of the angles of five pointed stars - How can you prove your observation?
I know the sum of 5 pointed stars is 180 degrees. Other than saying you just add up the interior angles, is there another way I can offer "proof"?? THANK YOU!!

2. sketch the circle that contains all 5 "points".

now use the fact that an angle inscribed in a circle equals one-half the measure of its intercepted arc.

3. THANKS!!!

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

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