Question

1. which describes how to construct an angle whose measure is equal to \\( \frac { 1 } { 4 } m \angle c a b \\)?

a. bisect \\( \angle c a b \\), then bisect the new angle.
b. bisect \\( \angle c a b \\), then bisect \\( \angle c a b \\) again.
c. bisect \\( \angle c a b \\), then copy the new angle.
d. copy \\( \angle c a b \\), then bisect the new angle.

Explanation:

Step1: Analyze the effect of bisecting an angle

When we bisect an angle \(\angle CAB\), we divide it into two equal parts. Let \(m\angle CAB = \theta\). After the first bisection, each of the new angles has a measure of \(\frac{\theta}{2}\).

Step2: Analyze the effect of bisecting the new - formed angle

If we then bisect the angle of measure \(\frac{\theta}{2}\), we divide it into two equal parts. Using the formula for bisection (dividing an angle into two equal sub - angles), if the angle to be bisected is \(\alpha=\frac{\theta}{2}\), after bisection, each sub - angle has a measure of \(\frac{\alpha}{2}\). Substituting \(\alpha = \frac{\theta}{2}\), we get \(\frac{\frac{\theta}{2}}{2}=\frac{\theta}{4}\).

Step3: Analyze other options

• Option B: Bisecting \(\angle CAB\) twice (but not using the first - bisected angle for the second bisection in the correct way). If we bisect \(\angle CAB\) (get two angles of measure \(\frac{\theta}{2}\)) and then bisect \(\angle CAB\) again (not using the previously bisected angles), we do not get \(\frac{\theta}{4}\).
• Option C: Bisecting \(\angle CAB\) (get \(\frac{\theta}{2}\)) and then copying \(\frac{\theta}{2}\) does not give \(\frac{\theta}{4}\).
• Option D: Copying \(\angle CAB\) (get \(\theta\)) and then bisecting gives \(\frac{\theta}{2}\), not \(\frac{\theta}{4}\).

Answer:

A. Bisect \(\angle CAB\), then bisect the new angle.