Question

find ( mangle 1 ) and ( mangle 2 ) given that lines ( n ) and ( m ) are parallel. justify your answer. ( mangle 1=square^{circ} ) by the alternate exterior angles theorem. corresponding angles postulate. same - side interior angles theorem. alternate interior angles theorem. vertical angles theorem.

Explanation:

Step1: Find \(m\angle1\)

By the corresponding angles postulate, when two parallel lines are cut by a transversal, corresponding angles are congruent. Here, \(\angle1\) and the angle that is vertically opposite to the \(75^{\circ}\) angle are corresponding angles. Since vertical angles are congruent (vertical angles theorem), and using the corresponding angles postulate for parallel lines \(n\) and \(m\) cut by the transversal, \(m\angle1 = 75^{\circ}\).

Step2: Find \(m\angle2\)

\(\angle2\) and the \(75^{\circ}\) angle are same - side interior angles. By the same - side interior angles theorem, if two parallel lines are cut by a transversal, then the sum of same - side interior angles is \(180^{\circ}\). So \(m\angle2=180^{\circ}- 75^{\circ}=105^{\circ}\).

Answer:

\(m\angle1 = 75^{\circ}\) by the corresponding angles postulate. \(m\angle2 = 105^{\circ}\) by the same - side interior angles theorem.