Question

1. what are the angles of rotation for a 20-gon? how many lines of symmetry (lines of reflection) will it have?

Explanation:

Step1: Find the angle of rotation for a regular n - gon

For a regular \(n\) - gon, the angle of rotation (the smallest angle by which the figure can be rotated to coincide with itself) is given by the formula \(\theta=\frac{360^{\circ}}{n}\), where \(n\) is the number of sides. For a 20 - gon, \(n = 20\). So, \(\theta=\frac{360^{\circ}}{20}=18^{\circ}\). The angles of rotation are all positive integer multiples of \(18^{\circ}\) that are less than \(360^{\circ}\), i.e., \(18^{\circ},36^{\circ},54^{\circ},\cdots,342^{\circ}\).

Step2: Find the number of lines of symmetry for a regular n - gon

A regular \(n\) - gon has \(n\) lines of symmetry. For a 20 - gon, since \(n = 20\), the number of lines of symmetry is 20.

Answer:

The angles of rotation are \(18^{\circ}k\) where \(k = 1,2,\cdots,19\) (i.e., \(18^{\circ},36^{\circ},\cdots,342^{\circ}\)) and the number of lines of symmetry is 20.