Question

what is m∠1?
m∠1 = 45 °
what are the remaining angle measures?
m∠2 = □°
m∠3 = ? °
m∠4 = ? °

Explanation:

Step1: Analyze ∠2

The lines are parallel (implied by corresponding angles), and the angle adjacent to 45° for ∠2 forms a linear pair or is vertical/alternate. Since the upper intersection has 45° and 135°, ∠2 should be equal to the vertical angle of 135°? Wait, no, looking at the diagram, the angle marked 45° at the lower intersection: ∠2 and the 45° angle? Wait, no, let's see: the two transversals? Wait, actually, the lines are parallel, so corresponding angles. The upper angle has 45° and 135°, so the lower intersection: ∠2 is supplementary to 45°? Wait, no, linear pair: ∠2 and 45° form a linear pair? Wait, no, the angle next to ∠2 is 45°, so ∠2 + 45° = 180°? Wait, no, maybe vertical angles or alternate interior. Wait, the upper intersection has angles 45°, 135°, 45°, 135°. The lower intersection: the angle labeled 45° is corresponding to the upper 45°, so ∠2 is vertical to the 135° angle? Wait, let's re-express:

At the upper intersection, two lines cross, forming angles 45°, 135°, 45°, 135°. At the lower intersection, the same transversal crosses the parallel line, so corresponding angles. So the angle adjacent to ∠2 (the 45° angle) is corresponding to the upper 45°, so ∠2 and that 45° angle are supplementary? Wait, no, linear pair: ∠2 + 45° = 180°? Then ∠2 = 135°? Wait, no, maybe I got it wrong. Wait, the diagram: the lower intersection has a 45° angle, then ∠2 is opposite? Wait, no, let's think about vertical angles and linear pairs.

Step2: Calculate ∠2

The angle marked 45° at the lower intersection and ∠2 form a linear pair (they are adjacent and form a straight line), so \( m\angle 2 + 45^\circ = 180^\circ \). So \( m\angle 2 = 180^\circ - 45^\circ = 135^\circ \).

Step3: Calculate ∠3

∠3 is vertical to the 45° angle (opposite angles when two lines cross), so \( m\angle 3 = 45^\circ \) (vertical angles are equal).

Step4: Calculate ∠4

∠4 is vertical to ∠2, so \( m\angle 4 = m\angle 2 = 135^\circ \) (vertical angles are equal). Or ∠4 is supplementary to ∠3: \( 180^\circ - 45^\circ = 135^\circ \).

Answer:

\( m\angle 2 = 135^\circ \), \( m\angle 3 = 45^\circ \), \( m\angle 4 = 135^\circ \)