Question

1. what are the angles of rotation for a 15-gon? how many line of symmetry (lines of reflection) will it have?

Explanation:

Step1: Find rotation angle

A regular \( n \)-gon has a rotational symmetry angle of \( \frac{360^\circ}{n} \). For a 15 - gon, \( n = 15 \), so the rotation angle is \( \frac{360^\circ}{15}=24^\circ \). The angles of rotation are \( 24^\circ k \) where \( k = 1,2,\cdots,14 \) (and also \( 360^\circ \) which is equivalent to \( 0^\circ \)).

Step2: Find lines of symmetry

A regular \( n \)-gon has \( n \) lines of symmetry. For a 15 - gon, \( n = 15 \), so it has 15 lines of symmetry.

Answer:

The angles of rotation are \( 24^\circ, 48^\circ, \cdots, 336^\circ \) (multiples of \( 24^\circ \) up to \( 336^\circ \)). The number of lines of symmetry is 15.