Given angle A, how can you construct an angle whose measure is 1/4 m of angle A?

Solution:

Detailed steps:

Given angle A, our objective is to construct an angle whose measure equals one-fourth of angle A.

This means dividing angle A into four equal parts, where each part corresponds to the required angle.

To achieve this, first construct the bisector of angle A, splitting it into two equal angles each measuring half of A. Then bisect one of these halves to create an angle that is one-quarter of A.

The resulting smaller angle is the desired one.

Thus, option A is correct: first bisect angle A, then bisect one of the resulting smaller angles.

Suppose angle A measures 80 degrees. Halving it would yield 40 degrees, and bisecting that angle again produces 20 degrees. This 20 degrees angle is exactly one-quarter of 80 degrees, since 20 times 4 equals 80.