Question

(01.02 lc)
beck used a compass and straightedge to accurately construct line segment os, as shown in the figure:
which could be the measures of angle pos and angle poq?
○ a $m\angle pos = 60^{\circ}, m\angle poq = 120^{\circ}$
○ b $m\angle pos = 50^{\circ}, m\angle poq = 110^{\circ}$
○ c $m\angle pos = 50^{\circ}, m\angle poq = 120^{\circ}$
○ d $m\angle pos = 60^{\circ}, m\angle poq = 110^{\circ}$

Explanation:

Step1: Recall the property of angle bisector

When a compass and straight - edge are used to construct a ray (here \(OS\)) from a vertex (\(O\)) of an angle (\(\angle POQ\)), if \(OS\) is an angle bisector, then \(\angle POQ = 2\angle POS\).

Step2: Check each option

• Option a:

If \(\angle POS=60^{\circ}\) and \(\angle POQ = 120^{\circ}\), since \(120^{\circ}=2\times60^{\circ}\), this satisfies the angle - bisector property.

• Option b:

If \(\angle POS = 50^{\circ}\), then \(2\angle POS=100^{\circ}
eq110^{\circ}\).

• Option c:

If \(\angle POS = 50^{\circ}\), then \(2\angle POS = 100^{\circ}
eq120^{\circ}\).

• Option d:

If \(\angle POS=60^{\circ}\), then \(2\angle POS = 120^{\circ}
eq110^{\circ}\).

Answer:

A. \(m\angle POS = 60^{\circ},m\angle POQ = 120^{\circ}\)