Question

given lines a and b are parallel, and m<1 = 50°. find m<4.
options: 30°, 180°, 50°, 130°

Explanation:

Step1: Identify Angle Relationships

Since lines \(a\) and \(b\) are parallel, and considering the transversal, \(\angle 1\) and \(\angle 4\) are corresponding angles (or we can use alternate - interior angles logic too). Corresponding angles formed by a transversal with parallel lines are equal. Also, we can check the vertical angles and alternate angles. First, \(\angle 1\) and \(\angle 2\) are vertical angles? No, wait, \(\angle 1\) and \(\angle 2\) are actually alternate - interior? Wait, no, let's look at the diagram. The angle \(\angle 1\) and \(\angle 4\): since \(a\parallel b\), and the transversal cuts them, \(\angle 1\) and \(\angle 4\) should be equal because they are corresponding angles (or we can see that \(\angle 1=\angle 2\) (vertical angles? No, \(\angle 1\) and \(\angle 2\) are actually alternate - interior? Wait, maybe a better way: \(\angle 1\) and \(\angle 3\) are same - side? No, wait, \(\angle 1\) and \(\angle 4\): since \(a\parallel b\), the angle \(\angle 1\) and \(\angle 4\) are equal because they are corresponding angles. Wait, actually, \(\angle 1\) and \(\angle 2\) are vertical angles? No, \(\angle 1\) and \(\angle 2\) are alternate - interior? Wait, maybe I made a mistake. Wait, the measure of \(\angle 1 = 50^{\circ}\). Then, \(\angle 1\) and \(\angle 4\): since \(a\parallel b\), and the transversal, \(\angle 1\) and \(\angle 4\) are equal. Wait, let's think again. The angle \(\angle 1\) and \(\angle 2\) are vertical angles? No, \(\angle 1\) and \(\angle 2\) are actually alternate - interior? Wait, no, in the diagram, \(\angle 1\) and \(\angle 4\): if we consider the two parallel lines \(a\) and \(b\) and the transversal, \(\angle 1\) and \(\angle 4\) are corresponding angles. So corresponding angles are equal when lines are parallel. So \(m\angle 1=m\angle 4\) because \(a\parallel b\) and the transversal creates corresponding angles.

Step2: Calculate \(m\angle 4\)

We know that \(m\angle 1 = 50^{\circ}\). Since \(\angle 1\) and \(\angle 4\) are equal (corresponding angles, parallel lines), then \(m\angle 4=m\angle 1 = 50^{\circ}\).

Answer:

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