Question

lines pq and rs are parallel. what is the measure of angle f in the figure below? a 117 b 17 c 73 d 107

Explanation:

Step1: Identify supplementary angles

The angle of \(73^\circ\) and its adjacent angle on line \(PQ\) are supplementary (they form a linear pair). So, the adjacent angle \( \angle BAC\) (or the corresponding angle) is \(180^\circ - 73^\circ = 107^\circ\)? Wait, no, first, let's recall parallel lines and transversals. The transversal cuts \(PQ\) and \(RS\). The angle at \(A\) ( \(73^\circ\)) and the angle corresponding to \( \angle F\)'s supplementary? Wait, no, let's find the consecutive interior angle or the supplementary angle. Wait, the angle adjacent to \(73^\circ\) on the straight line \(PQ\) (since \(PQ\) is a straight line) is \(180^\circ - 73^\circ = 107^\circ\). Now, since \(PQ \parallel RS\), the corresponding angle (or alternate interior/exterior) to this \(107^\circ\) angle will be equal to \( \angle F\)'s supplementary? Wait, no, actually, the angle at \(A\) ( \(73^\circ\)) and the angle that is vertical or corresponding. Wait, maybe better: the angle adjacent to \(73^\circ\) (linear pair) is \(180 - 73 = 107^\circ\). Then, since \(PQ \parallel RS\), the angle \( \angle F\) and this \(107^\circ\) angle? Wait, no, the transversal creates corresponding angles. Wait, the angle at \(A\) ( \(73^\circ\)) and the angle at \(D\) (or \(G\)): since \(PQ \parallel RS\), the consecutive interior angles? No, maybe the angle \( \angle F\) is supplementary to the angle that is equal to \(73^\circ\). Wait, let's correct: the angle of \(73^\circ\) and the angle we need ( \( \angle F\)): since \(PQ \parallel RS\), and the transversal, the angle adjacent to \(73^\circ\) (linear pair) is \(107^\circ\), and since \(PQ \parallel RS\), the corresponding angle to this \(107^\circ\) angle is equal to \( \angle F\)? Wait, no, \( \angle F\) and the angle that is \(107^\circ\) are corresponding? Wait, maybe the angle \( \angle F\) is equal to \(180^\circ - 73^\circ = 107^\circ\) because they are same - side interior angles? Wait, no, same - side interior angles are supplementary. Wait, the angle of \(73^\circ\) and \( \angle F\): if \(PQ \parallel RS\), then the angle adjacent to \(73^\circ\) (linear pair, \(107^\circ\)) and \( \angle F\) are corresponding angles, so \( \angle F = 107^\circ\).

Step2: Calculate the measure of \( \angle F\)

We know that a straight line forms a \(180^\circ\) angle. So, the angle supplementary to \(73^\circ\) is \(180^\circ - 73^\circ=\ 107^\circ\). Since \(PQ\parallel RS\), by the properties of parallel lines and transversals (corresponding angles or alternate interior angles, or consecutive interior angles), the measure of \( \angle F\) is equal to this supplementary angle, which is \(107^\circ\).

Answer:

D. \(107^\circ\)