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 vertical angles - but since lines are parallel, alternate interior angles are equal; also, if we consider the linear pair or other, but here, looking at the diagram, \(\angle 1\) and \(\angle 4\) should be supplementary? Wait, no, wait. Wait, maybe I made a mistake. Wait, no, let's re-examine. Wait, lines \(a\) and \(b\) are parallel, transversal \(f\). Wait, \(\angle 1\) and the angle adjacent to \(\angle 4\) - no, wait, actually, \(\angle 1\) and \(\angle 4\): let's see, if we look at the diagram, \(\angle 1\) and \(\angle 4\) are same - side? No, wait, maybe they are supplementary? Wait, no, wait, the sum of \(\angle 1\) and \(\angle 4\) should be 180? Wait, no, maybe I messed up. Wait, no, let's think again. Wait, when two parallel lines are cut by a transversal, consecutive interior angles are supplementary. Wait, but \(\angle 1\) and \(\angle 4\): let's see, \(\angle 1\) is 110 degrees. Then, the angle adjacent to \(\angle 4\) (let's say \(\angle x\)) would be equal to \(\angle 1\) because they are corresponding angles. Then \(\angle x + \angle 4 = 180\) (linear pair). Wait, no, maybe \(\angle 1\) and \(\angle 4\) are supplementary? Wait, no, let's calculate. If \(m\angle 1 = 110^\circ\), then the angle supplementary to \(\angle 1\) is \(180 - 110 = 70^\circ\). Wait, but maybe \(\angle 4\) is equal to that supplementary angle? Wait, no, maybe I got the diagram wrong. Wait, the options include 70 and 110. Wait, maybe \(\angle 1\) and \(\angle 4\) are same - side interior angles? No, same - side interior angles are supplementary. Wait, no, maybe they are alternate exterior angles? No, alternate exterior angles are equal. Wait, maybe the diagram shows that \(\angle 1\) and \(\angle 4\) are supplementary. Wait, let's check the options. If \(m\angle 1 = 110^\circ\), then \(m\angle 4 = 180 - 110 = 70^\circ\). Because they are same - side interior angles (consecutive interior angles) which are supplementary when lines are parallel. So:

Step1: Recall consecutive interior angles theorem

When two parallel lines are cut by a transversal, consecutive interior angles are supplementary (sum to \(180^\circ\)).

Step2: Calculate \(m\angle 4\)

Given \(m\angle 1 = 110^\circ\), and \(\angle 1\) and \(\angle 4\) are consecutive interior angles (supplementary), so:
\(m\angle 1 + m\angle 4 = 180^\circ\)
\(110^\circ + m\angle 4 = 180^\circ\)
\(m\angle 4 = 180^\circ - 110^\circ = 70^\circ\)

Answer:

\(70^\circ\) (corresponding to the option with \(70^\circ\))