Question

given below, \\( \overline{ef} \\) cuts across two parallel segments \\( \overline{ab} \\) and \\( \overline{cd} \\). if the measure of \\( 0^{\circ} \\), then what is the measure of \\( \angle epb \\)? select one answer a \\( 40^{\circ} \\) b \\( 70^{\circ} \\) c \\( 110^{\circ} \\) d \\( 250^{\circ} \\)

Explanation:

Step1: Identify Angle Relationship

Since \( AB \parallel CD \) and \( EF \) is a transversal, \( \angle EPB \) and the \( 110^\circ \) angle at \( Q \) are same - side interior angles? Wait, no, actually, the angle adjacent to the \( 110^\circ \) angle (linear pair) and \( \angle EPB \) are corresponding angles. First, find the linear pair of the \( 110^\circ \) angle. The linear pair of \( 110^\circ \) is \( 180^\circ - 110^\circ=70^\circ \)? Wait, no, wait. Wait, \( AB \parallel CD \), and \( EF \) is a transversal. The angle at \( Q \) is \( 110^\circ \), and \( \angle EPB \) and the angle supplementary to \( 110^\circ \) (since they are same - side interior angles? No, actually, \( \angle EPB \) and the angle that is vertical or corresponding. Wait, let's correct. The angle at \( Q \) (let's call it \( \angle DQF = 110^\circ \)), then the consecutive interior angle to \( \angle DQF \) (along \( CD \)) is \( 180 - 110=70^\circ \)? No, wait, \( AB \parallel CD \), so \( \angle EPB \) and the angle that is equal to the supplementary angle of \( 110^\circ \)? Wait, no. Let's think again. The angle at \( Q \) (let's say \( \angle CQE \) is supplementary to \( 110^\circ \)? Wait, the diagram: \( AB \) and \( CD \) are parallel, \( EF \) intersects them at \( P \) and \( Q \). The angle at \( Q \) (below \( CD \)) is \( 110^\circ \), so the angle above \( CD \) at \( Q \) (let's call it \( \angle CQE \)) is \( 180 - 110 = 70^\circ \). Then, since \( AB \parallel CD \), \( \angle EPB \) is equal to \( \angle CQE \) (corresponding angles). So \( \angle EPB=70^\circ \)? Wait, no, wait. Wait, the angle at \( Q \) is \( 110^\circ \), and \( \angle EPB \) and the \( 110^\circ \) angle: are they same - side interior angles? If \( AB \parallel CD \), then same - side interior angles are supplementary. So \( \angle EPB+110^\circ = 180^\circ \)? No, that would be if they are same - side interior angles. Wait, let's check the positions. \( AB \) is above, \( CD \) is below. \( EF \) is a transversal going from \( E \) (top) to \( F \) (bottom). So \( \angle EPB \) is at \( P \) on \( AB \), and the angle at \( Q \) on \( CD \) is \( 110^\circ \) (below \( CD \)). So the same - side interior angles would be \( \angle EPB \) and the angle above \( CD \) at \( Q \). Wait, the angle above \( CD \) at \( Q \) is \( 180 - 110 = 70^\circ \). Then, since \( AB \parallel CD \), same - side interior angles are supplementary? No, same - side interior angles are supplementary. Wait, no, \( \angle EPB \) and the angle above \( CD \) at \( Q \): if \( AB \parallel CD \), then \( \angle EPB+\) (angle above \( CD \) at \( Q\)) \( = 180^\circ \)? No, that can't be. Wait, maybe I got the direction wrong. Let's use the consecutive interior angles theorem. Consecutive interior angles are supplementary. So if \( AB \parallel CD \), and \( EF \) is a transversal, then \( \angle EPB \) and the angle that is consecutive interior to it (at \( Q \)) are supplementary. The angle at \( Q \) (below \( CD \)) is \( 110^\circ \), so the consecutive interior angle to \( \angle EPB \) is \( 110^\circ \), so \( \angle EPB + 110^\circ=180^\circ \), so \( \angle EPB = 180 - 110=70^\circ \). Yes, that makes sense. Because consecutive interior angles (same - side interior angles) are supplementary when lines are parallel. So \( \angle EPB \) and the \( 110^\circ \) angle are same - side interior angles, so they add up to \( 180^\circ \). So \( \angle EPB=180 - 110 = 70^\circ \).

Answer:

B. \( 70^\circ \)