Question

which set of equations is enough information to prove that lines a and b are parallel lines cut by transversal f? m∠4 = 110° and m∠3 = 70° m∠1 = 110° and m∠2 = 110° m∠1 = 110° and m∠3 = 70° m∠2 = 110° and m∠3 = 110°

Explanation:

Step1: Recall Parallel Line Theorems

To prove lines \(a\) and \(b\) are parallel, we use angle relationships (e.g., supplementary angles, alternate interior, corresponding, etc.). If two angles are supplementary (sum to \(180^\circ\)) and are same - side interior angles, or if alternate interior angles are equal, etc., lines are parallel.

Step2: Analyze Option 1 (\(m\angle4 = 110^\circ\) and \(m\angle3 = 70^\circ\))

Check if \(\angle4\) and \(\angle3\) are supplementary. \(110^\circ+70^\circ = 180^\circ\). But \(\angle4\) and \(\angle3\) are adjacent angles (forming a linear pair), not related to parallel lines (they are on the same line \(b\) with transversal \(f\)). So this doesn't prove \(a\parallel b\).

Step3: Analyze Option 2 (\(m\angle1 = 110^\circ\) and \(m\angle2 = 110^\circ\))

\(\angle1\) and \(\angle2\) are adjacent angles (linear pair? No, they are vertical or adjacent? Wait, \(\angle1\) and \(\angle2\) are adjacent and form a linear pair? Wait, no, in the diagram, \(\angle1\) and \(\angle2\) are adjacent angles. If \(m\angle1=m\angle2 = 110^\circ\), their sum is \(220^\circ eq180^\circ\), so they can't be a linear pair. Also, they don't give a relationship for parallel lines.

Step4: Analyze Option 3 (\(m\angle1 = 110^\circ\) and \(m\angle3 = 70^\circ\))

\(\angle1\) and \(\angle2\) are vertical angles? Wait, \(\angle1\) and \(\angle2\) are adjacent, but \(\angle1\) and \(\angle3\): \(\angle1\) and \(\angle2\) are adjacent, \(\angle2\) and \(\angle3\) are adjacent. Wait, \(\angle1\) and \(\angle3\): Let's see the positions. \(\angle1\) and \(\angle3\) are same - side interior angles? Wait, \(m\angle1 = 110^\circ\), \(m\angle3=70^\circ\), and \(110^\circ + 70^\circ=180^\circ\). \(\angle1\) and \(\angle3\) are same - side interior angles (since \(a\) and \(b\) are cut by transversal \(f\), \(\angle1\) is above \(a\), \(\angle3\) is below \(a\) on the same side of transversal \(f\)). If same - side interior angles are supplementary, then lines are parallel. Wait, no, wait: Wait, \(\angle1\) and \(\angle3\) - Wait, maybe I made a mistake. Wait, let's re - examine. Wait, \(\angle1\) and \(\angle2\) are vertical angles? No, \(\angle1\) and \(\angle3\): Let's look at the alternate interior angles or same - side. Wait, actually, \(\angle1\) and \(\angle3\): If \(m\angle1 = 110^\circ\) and \(m\angle3 = 70^\circ\), and \(\angle1+\angle3=180^\circ\), and \(\angle1\) and \(\angle3\) are same - side interior angles (since \(a\) and \(b\) are cut by \(f\)), then by the same - side interior angles supplementary theorem, \(a\parallel b\). Wait, no, wait, let's check the other options.

Step5: Analyze Option 4 (\(m\angle2 = 110^\circ\) and \(m\angle3 = 110^\circ\))

\(\angle2\) and \(\angle3\) are alternate interior angles (if \(a\parallel b\), alternate interior angles should be equal). But \(m\angle2 = 110^\circ\) and \(m\angle3 = 110^\circ\)? Wait, no, if \(a\parallel b\), alternate interior angles \(\angle2\) and \(\angle3\) should be equal? Wait, no, in the diagram, \(\angle2\) and \(\angle3\) are alternate interior angles? Wait, no, \(\angle2\) is on line \(a\), \(\angle3\) is on line \(b\), cut by transversal \(f\). If \(m\angle2=m\angle3 = 110^\circ\), but alternate interior angles being equal would prove parallel, but wait, \(\angle2\) and \(\angle3\) - if they are equal, but in this case, if \(m\angle2 = 110^\circ\) and \(m\angle3 = 110^\circ\), but let's check the first option again. Wait, no, let's go back to option 3. Wait, \(\angle1\) and \(\angle3\): Wait, \(\angle1\) and \(\angle2\) are adjacent, \(\angle2\) and \(\angle3\) are adjacent. Wait, \(\angle1\) and \(\angle3\) are same - side interior angles. The sum of same - side interior angles should be \(180^\circ\) for lines to be parallel. \(110^\circ + 70^\circ=180^\circ\), so this satisfies the same - side interior angles supplementary condition, proving \(a\parallel b\).

Wait, no, wait option 1: \(\angle4\) and \(\angle3\) are supplementary, but they are on line \(b\). Option 3: \(\angle1\) (above \(a\)) and \(\angle3\) (below \(b\)) - wait, no, let's re - draw mentally. Lines \(a\) and \(b\) are cut by transversal \(f\). \(\angle1\) and \(\angle3\): \(\angle1\) is at the intersection of \(a\) and \(f\), \(\angle3\) is at the intersection of \(b\) and \(f\). So \(\angle1\) and \(\angle3\) are same - side interior angles. If their sum is \(180^\circ\) (\(110 + 70=180\)), then \(a\parallel b\) by the same - side interior angles supplementary theorem.

Answer:

\(m\angle1 = 110^\circ\) and \(m\angle3 = 70^\circ\) (the third option)