Question

1. given: $p\parallel q$; $m\parallel n$; $m\angle1 = 75^{circ}$ what is $m\angle2$? a. $15^{circ}$ b. $75^{circ}$ c. $90^{circ}$ d. $105^{circ}$

Explanation:

Step1: Use the property of parallel lines

Since \(p\parallel q\) and \(m\parallel n\), the quadrilateral formed is a parallelogram. In a parallelogram, opposite angles are equal. But here, we can also use the property of alternate - interior angles.
Let's first find the angle adjacent to \(\angle1\). The angle adjacent to \(\angle1\) (let's call it \(\angle a\)) forms a linear pair with \(\angle1\). So \(m\angle a = 180^{\circ}-m\angle1\).

$$m\angle a=180 - 75=105^{\circ}$$

Step2: Use the property of parallelogram (or parallel - line properties)

Because \(p\parallel q\) and \(m\parallel n\), \(\angle2\) and the angle adjacent to \(\angle1\) (\(\angle a\)) are equal (by the property of a parallelogram: opposite angles of a parallelogram are equal. Also, when two pairs of parallel lines \(p\parallel q\) and \(m\parallel n\) intersect, we can consider the translation of angles. Another way: if we consider the transversal cutting the parallel lines \(m\) and \(n\) and \(p\) and \(q\), \(\angle2\) and the angle adjacent to \(\angle1\) are congruent as they are corresponding angles for the parallelogram - like structure formed by the two pairs of parallel lines)

Answer:

D. \(105^{\circ}\)