Question

which of the following transformations carry this regular polygon onto itself?
reflection across l
rotation of 36° clockwise
rotation of 72° counterclockwise
rotation of 72° clockwise

Explanation:

Step1: Properties of regular pentagon

A regular pentagon has 5 sides. The angle of rotational symmetry is $\frac{360^{\circ}}{n}$, where $n = 5$. So, $\frac{360^{\circ}}{5}=72^{\circ}$.

Step2: Reflection property

A line of symmetry (like line $l$) for a regular pentagon (if it is an axis of symmetry) will map the pentagon onto itself.

Step3: Rotation property

Rotations of multiples of $72^{\circ}$ (clock - wise or counter - clockwise) will map a regular pentagon onto itself. A rotation of $36^{\circ}$ (which is not a multiple of $72^{\circ}$) will not map the regular pentagon onto itself.

Answer:

reflection across $l$, rotation of $72^{\circ}$ counterclockwise, rotation of $72^{\circ}$ clockwise