Question

given lines a and b are parallel, and m<1 = 50°. find m<3.
options: 50°, 70°, 110°, 180°

Explanation:

Step1: Identify Angle Relationship

Lines \(a\) and \(b\) are parallel, and the transversal creates corresponding angles or check for same - side interior? Wait, actually, \(\angle1\) and \(\angle3\): Wait, no, first, \(\angle1\) and \(\angle2\) are alternate interior? Wait, no, let's look at the diagram. The two parallel lines \(a\) and \(b\), and the transversal (the horizontal line) and the other transversal (the lines \(a\) and \(b\) are the slant lines? Wait, no, in the diagram, lines \(a\) and \(b\) are parallel (the slant upward arrows), and the horizontal line is a transversal, and another slant downward line? Wait, no, the angles: \(\angle1\) and \(\angle3\) - Wait, actually, \(\angle1\) and \(\angle3\) are same - side? No, wait, \(\angle1\) and \(\angle2\) are alternate interior angles, so \(m\angle1=m\angle2 = 50^{\circ}\). Then \(\angle2\) and \(\angle3\) are same - side interior angles? Wait, no, \(\angle2\) and \(\angle3\) are supplementary? Wait, no, let's re - examine. Wait, the two parallel lines \(a\) and \(b\), and the horizontal line is a transversal. The angle \(\angle1\) and \(\angle3\): Wait, actually, \(\angle1\) and \(\angle3\) are corresponding angles? Wait, no, maybe \(\angle1\) and \(\angle3\) are same - side? Wait, no, let's think again. The lines \(a\) and \(b\) are parallel, and the transversal (the horizontal line) and the two slant lines ( \(a\) and \(b\) ) form angles. Wait, \(\angle1\) and \(\angle3\): Wait, no, \(\angle1\) and \(\angle3\) are actually same - side interior? No, wait, \(\angle1\) and \(\angle3\) are supplementary? Wait, no, I think I made a mistake. Wait, \(\angle1\) and \(\angle3\): Let's see, the angle \(\angle1\) and \(\angle3\) - if we consider the two parallel lines \(a\) and \(b\), and the transversal (the horizontal line), then \(\angle1\) and \(\angle3\) are same - side interior angles? Wait, no, \(\angle1\) and \(\angle3\) are actually supplementary? Wait, no, let's use the fact that \(\angle1\) and \(\angle2\) are alternate interior (so \(m\angle1 = m\angle2=50^{\circ}\)), and \(\angle2\) and \(\angle3\) are supplementary (since they are same - side interior angles between parallel lines), so \(m\angle2 + m\angle3=180^{\circ}\). Wait, no, that can't be. Wait, maybe \(\angle1\) and \(\angle3\) are corresponding angles. Wait, the diagram: line \(a\) and line \(b\) are parallel, the horizontal line is a transversal, and the two slant lines ( \(a\) and \(b\) ) are cut by the horizontal transversal. So \(\angle1\) and \(\angle3\) are corresponding angles, so \(m\angle1=m\angle3\)? Wait, no, that would be if the transversal is the same. Wait, maybe the two slant lines ( \(a\) and \(b\) ) are parallel, and the horizontal line and the other slant line (the downward one) are transversals. Wait, I think I messed up the diagram. Let's re - interpret: The two parallel lines are \(a\) (upward slant) and \(b\) (upward slant), and the horizontal line is a transversal, and the downward slant line is another transversal. The angle \(\angle1\) is between line \(a\) and the horizontal line, and \(\angle3\) is between line \(b\) and the horizontal line. Since \(a\parallel b\), \(\angle1\) and \(\angle3\) are same - side? No, wait, \(\angle1\) and \(\angle3\) are actually corresponding angles, so they should be equal? But that would mean \(m\angle3 = 50^{\circ}\), but that contradicts the supplementary idea. Wait, no, maybe the diagram has \(\angle1\) and \(\angle3\) as same - side interior angles. Wait, no, let's look at the answer options. The options are \(50^{\circ}\), \(70^{\circ}\), \(110^{\circ}\), \(180^{\circ}\). If \(\angle1\) and \(\angle3\) are supplementary, then \(m\angle3=180 - 50=130\), which is not an option. Wait, I must have misidentified the angles. Wait, maybe \(\angle1\) and \(\angle3\) are alternate exterior? No. Wait, maybe the two parallel lines are the slant lines ( \(a\) and \(b\) ), and the horizontal line is a transversal, and the angle \(\angle1\) and \(\angle3\) are corresponding angles. So if \(a\parallel b\), then corresponding angles are equal. So \(m\angle1=m\angle3 = 50^{\circ}\). That must be it. Because the lines \(a\) and \(b\) are parallel, and the transversal (the horizontal line) cuts them, so \(\angle1\) and \(\angle3\) are corresponding angles, so they are equal.

Step2: Calculate \(m\angle3\)

Since \(\angle1\) and \(\angle3\) are corresponding angles (because \(a\parallel b\) and the transversal creates corresponding angles), and \(m\angle1 = 50^{\circ}\), then \(m\angle3=m\angle1 = 50^{\circ}\).

Answer:

\(50^{\circ}\) (the option with \(50^{\circ}\))