Question

applying angle pair relationships
what is ( mangle 2 )? (square^circ)
what is ( mangle 1 )? (square^circ)

Explanation:

Step1: Find \( m\angle 2 \)

Angles \( \angle 2 \) and the \( 75^\circ \) angle are alternate interior angles (since the lines are parallel, as indicated by the arrow marks). Alternate interior angles are equal. Wait, no, actually, \( \angle 3 \) and the \( 75^\circ \) angle are corresponding? Wait, no, let's look at the transversal. Wait, the angle labeled \( 75^\circ \) and \( \angle 3 \): no, \( \angle 2 \) and the angle adjacent to \( 75^\circ \)? Wait, no, \( \angle 2 \) and \( \angle 3 \) are adjacent, and \( \angle 3 \) and the \( 75^\circ \) angle: wait, the two horizontal lines (x and w) are parallel, and the line z is a transversal. Wait, \( \angle 3 \) and the \( 75^\circ \) angle: are they corresponding? Wait, no, \( \angle 2 \) and the angle with \( 75^\circ \): wait, \( \angle 2 \) and \( \angle 5 \) are vertical? No, \( \angle 2 \) and \( \angle 3 \) are supplementary? Wait, no, let's correct. The angle \( 75^\circ \) and \( \angle 3 \): since the two horizontal lines (x and w) are parallel, and line z is a transversal, \( \angle 3 \) and the \( 75^\circ \) angle are alternate interior angles, so \( m\angle 3 = 75^\circ \). Then, \( \angle 2 \) and \( \angle 3 \) are adjacent and form a linear pair? Wait, no, \( \angle 2 \) and \( \angle 3 \): wait, the vertical line? No, the horizontal line (the top one) and the transversal z: \( \angle 1 \) and \( \angle 4 \) are vertical, \( \angle 2 \) and \( \angle 3 \) are adjacent. Wait, actually, \( \angle 2 \) and the \( 75^\circ \) angle: are they corresponding? Wait, no, the two horizontal lines (x and w) are parallel, so the transversal z creates alternate interior angles. So \( \angle 2 \) and the angle that's \( 75^\circ \): wait, \( \angle 5 \) and the \( 75^\circ \) angle: no, \( \angle 3 \) and the \( 75^\circ \) angle are alternate interior, so \( m\angle 3 = 75^\circ \). Then, \( \angle 2 \) and \( \angle 3 \) are adjacent and form a linear pair? Wait, no, \( \angle 2 \) and \( \angle 3 \) are on a straight line? Wait, the top horizontal line: \( \angle 1 \), \( \angle 4 \), \( \angle 13 \), \( \angle 16 \) are on one line? No, the top horizontal line (x) and the middle horizontal line (the one with \( 13,16,14,15 \))? Wait, no, the two horizontal lines are x (with \( 13,16,14,15 \)) and w (with \( 9,12,10,11 \)), and the other horizontal line is the top one with \( 1,4,2,3 \)? Wait, maybe the two horizontal lines (the top one with \( 1,4,2,3 \) and the middle one with \( 5,7, \) and the \( 75^\circ \)) are parallel? Wait, the arrows on the horizontal lines indicate they are parallel. So the transversal is z. So \( \angle 2 \) and \( \angle 5 \) are corresponding? No, \( \angle 3 \) and the \( 75^\circ \) angle are alternate interior, so \( m\angle 3 = 75^\circ \). Then, \( \angle 2 \) and \( \angle 3 \) are adjacent and form a linear pair? Wait, no, \( \angle 2 \) and \( \angle 3 \) are vertical? No, \( \angle 1 \) and \( \angle 3 \) are vertical? Wait, maybe I made a mistake. Let's look at \( \angle 2 \) and the \( 75^\circ \) angle. Wait, \( \angle 2 \) and the angle labeled \( 75^\circ \): are they same - side interior? No, alternate interior. Wait, if the two horizontal lines are parallel, then \( \angle 2 \) and the \( 75^\circ \) angle are equal? Wait, no, \( \angle 3 \) and the \( 75^\circ \) angle are alternate interior, so \( m\angle 3 = 75^\circ \). Then, \( \angle 2 \) and \( \angle 3 \) are supplementary? Wait, no, \( \angle 2 \) and \( \angle 3 \) are adjacent and form a linear pair? Wait, the line z intersects the top horizontal line, creating \( \angle 1, \angle 4, \angle 2, \angle 3 \). So \( \angle 2 \) and \( \angle 3 \) are adjacent and form a linear pair, so they are supplementary? Wait, no, \( \angle 1 \) and \( \angle 2 \) are supplementary, \( \angle 3 \) and \( \angle 4 \) are supplementary. Wait, maybe \( \angle 2 \) and the \( 75^\circ \) angle are equal because they are corresponding angles. Wait, let's assume that the two horizontal lines (the top one with \( 1,4,2,3 \) and the middle one with \( 5,7, \) and \( 75^\circ \)) are parallel. Then, the transversal z creates corresponding angles. So \( \angle 2 \) corresponds to the angle that's \( 75^\circ \)? Wait, no, \( \angle 3 \) corresponds to the \( 75^\circ \) angle. So \( m\angle 3 = 75^\circ \). Then, \( \angle 2 \) and \( \angle 3 \) are vertical angles? No, \( \angle 1 \) and \( \angle 3 \) are vertical angles? Wait, \( \angle 1 \) and \( \angle 3 \) are vertical, so \( m\angle 1 = m\angle 3 = 75^\circ \)? No, that can't be. Wait, maybe \( \angle 2 \) and the \( 75^\circ \) angle are supplementary. Wait, let's start over.

We know that when two parallel lines are cut by a transversal, alternate interior angles are equal. Let's identify the parallel lines. The two horizontal lines (let's call them line A: with \( 1,4,2,3 \) and line B: with \( 5,7, \) and \( 75^\circ \)) are parallel (indicated by the arrow marks). The transversal is line z. So the angle \( 75^\circ \) and \( \angle 3 \) are alternate interior angles, so \( m\angle 3 = 75^\circ \). Now, \( \angle 2 \) and \( \angle 3 \) are adjacent and form a linear pair? Wait, no, \( \angle 2 \) and \( \angle 3 \) are on a straight line? Wait, line A is a straight line, so \( \angle 2 + \angle 3 = 180^\circ \)? No, that would be if they are supplementary, but if \( \angle 3 = 75^\circ \), then \( \angle 2 = 180 - 75 = 105^\circ \)? Wait, no, maybe \( \angle 2 \) and \( \angle 3 \) are vertical angles? No, vertical angles are opposite each other. \( \angle 1 \) and \( \angle 3 \) are vertical angles, so \( m\angle 1 = m\angle 3 = 75^\circ \), and \( \angle 2 \) and \( \angle 4 \) are vertical angles. Then, \( \angle 2 + \angle 3 = 180^\circ \) (linear pair), so \( m\angle 2 = 180 - 75 = 105^\circ \), and \( m\angle 1 = m\angle 3 = 75^\circ \).

Step2: Verify the angle relationships

- For \( m\angle 2 \): Since \( \angle 2 \) and \( \angle 3 \) form a linear pair (they are adjacent and on a straight line), their measures add up to \( 180^\circ \). We found \( m\angle 3 = 75^\circ \) (alternate interior angles with the \( 75^\circ \) angle), so \( m\angle 2 = 180 - 75 = 105^\circ \).
- For \( m\angle 1 \): \( \angle 1 \) and \( \angle 3 \) are vertical angles, so they are equal. Since \( m\angle 3 = 75^\circ \), \( m\angle 1 = 75^\circ \).

Answer:

\( m\angle 2 = \boxed{105}^\circ \), \( m\angle 1 = \boxed{75}^\circ \)