Question

lines a and b are parallel lines cut by transversal f. if m∠1 = 110°, what is m∠4? 20° 70° 110° 130°

Explanation:

Step1: Identify Angle Relationship

Lines \(a\) and \(b\) are parallel, cut by transversal \(f\). \(\angle 1\) and \(\angle 4\) are alternate interior angles (or can also be considered as corresponding/alternate exterior, but key is they are congruent due to parallel lines). Wait, no—wait, actually, let's check the diagram. Wait, maybe they are same - side? No, wait, no—wait, actually, \(\angle 1\) and the angle adjacent to \(\angle 4\) (let's say \(\angle x\)) would be corresponding, but maybe \(\angle 1\) and \(\angle 4\) are supplementary? Wait, no, let's re - examine.

Wait, lines \(a\) and \(b\) are parallel, transversal \(f\). \(\angle 1\) and the angle that is vertical or alternate? Wait, no, maybe \(\angle 1\) and \(\angle 4\) are same - side interior? No, wait, the sum of same - side interior angles is \(180^{\circ}\)? Wait, no, same - side interior angles are supplementary. Wait, maybe I made a mistake. Wait, let's look at the positions. \(\angle 1\) and \(\angle 4\): if we consider the parallel lines \(a\) and \(b\), and transversal \(f\), \(\angle 1\) and \(\angle 4\) are actually alternate interior angles? No, wait, no—wait, maybe \(\angle 1\) and the angle opposite to \(\angle 4\) (vertical angle) is equal to \(\angle 1\), but \(\angle 4\) and that angle—wait, no, let's think again.

Wait, the correct relationship: when two parallel lines are cut by a transversal, alternate interior angles are equal, corresponding angles are equal, and same - side interior angles are supplementary. Let's assume that \(\angle 1\) and \(\angle 4\) are same - side interior angles? No, wait, maybe \(\angle 1\) and the angle adjacent to \(\angle 4\) (let's call it \(\angle 5\)) is equal to \(\angle 1\) (corresponding angles). Then \(\angle 4\) and \(\angle 5\) are supplementary (linear pair). So if \(m\angle1 = 110^{\circ}\), then \(m\angle5=110^{\circ}\), and \(m\angle4 = 180^{\circ}- 110^{\circ}=70^{\circ}\).

Step2: Calculate \(m\angle4\)

Since \(\angle 1\) and \(\angle 4\) (along with their adjacent angles) form a linear - pair - related supplementary angle situation (because of parallel lines and transversal), we use the formula for supplementary angles: \(m\angle1 + m\angle4=180^{\circ}\) (wait, no, maybe \(\angle 1\) and the angle that is supplementary to \(\angle 4\) is equal to \(\angle 1\)). Wait, let's do it properly.

If two parallel lines are cut by a transversal, same - side interior angles are supplementary. Let's say \(\angle 1\) and \(\angle 4\) are same - side interior angles? No, maybe \(\angle 1\) and the angle adjacent to \(\angle 4\) (let's call it \(\angle y\)) is a corresponding angle to \(\angle 1\), so \(m\angle y=m\angle1 = 110^{\circ}\). Then \(\angle y\) and \(\angle 4\) are supplementary (they form a linear pair), so \(m\angle4=180^{\circ}-m\angle y = 180 - 110=70^{\circ}\).

Answer:

\(70^{\circ}\) (the option with \(70^{\circ}\))