Question

1. reflection across l

rotation of 40° clockwise
rotation of 30° counterclockwise
rotation of 45° counterclockwise

Explanation:

Step1: Analyze the polygon

The figure is a regular octagon. The central angle for a regular octagon is $\frac{360^\circ}{8} = 45^\circ$.

Step2: Check rotational symmetry

A regular octagon has rotational symmetry of order 8, meaning it maps onto itself when rotated by multiples of $45^\circ$ (since $360\div8 = 45$). Also, it has reflection symmetry across lines like \( l \) (as seen with the marked congruent segments). For rotation, a $45^\circ$ counterclockwise rotation (or clockwise, but $45^\circ$ is the key angle here) will map the octagon onto itself. The $40^\circ$ and $30^\circ$ rotations are not multiples of the central angle, so they won't map it onto itself. The $45^\circ$ counterclockwise rotation is a valid symmetry operation for the regular octagon. Also, reflection across \( l \) is valid (as the marked sides are congruent across \( l \)). But if we consider the rotation options, the correct rotational symmetry angle is $45^\circ$ counterclockwise.

Answer:

The correct options are:

• reflection across \( l \)
• rotation of \( 45^\circ \) counterclockwise