First prove that the line segments are chords of a circle. Then
prove that the angle of turn between one chord and the next is
equal to the angle subtended by the chord at the centre of the
circle.
You do not need to have met the Logo programming language
previously in order to do this question. You can download the free
MSWlogo program from http://www.softronix.com/logo.html
If you have not used Logo before then, from the first part of this
simple introduction to Logo for
beginners , you can learn enough in five minutes to use Logo
to experiment in drawing the spirostars.