Question

in the figure, line m is parallel to line n, and line q is perpendicular to line p. suppose m∠1 = 40°. find m∠7.

Explanation:

Step1: Identify Vertical Angles

∠1 and ∠3 are vertical angles, so \( m\angle 3 = m\angle 1 = 40^\circ \).

Step2: Use Parallel Lines and Transversal

Since line \( m \parallel n \), and line \( p \) is a transversal, ∠3 and ∠7 are alternate interior angles? Wait, no, line \( q \) is perpendicular to \( p \), and there's a right angle (∠5 is right? Wait, the figure has a right angle at ∠5, so ∠3 + ∠7 + 90°? Wait, no, let's re-examine. Wait, line \( q \) is perpendicular to \( p \), so ∠3 + ∠7 = 90°? Wait, no, the right angle is at ∠5, so ∠4 + ∠5 + ∠6? Wait, maybe better: ∠1 and ∠3 are vertical (40°), line \( m \parallel n \), and line \( p \) (the transversal) with line \( q \) perpendicular to \( p \), so the angle between \( q \) and \( p \) is 90°. Wait, ∠3 + ∠7 = 90°? Wait, no, ∠3 and ∠7: since \( m \parallel n \), and \( q \) is a transversal? Wait, maybe ∠1 and ∠7: wait, ∠1 is 40°, line \( m \parallel n \), and \( q \) is perpendicular to \( p \), so the angle between \( q \) and \( m \) (or \( n \)): Wait, let's see, ∠1 and ∠3 are vertical (40°), then ∠3 + ∠7 = 90°? Wait, no, the right angle is at ∠5, so ∠4 + ∠5 + ∠6? Wait, maybe the correct approach: ∠1 and ∠3 are vertical (40°), line \( m \parallel n \), so ∠3 = ∠7? No, that can't be. Wait, no, line \( q \) is perpendicular to \( p \), so ∠3 + ∠7 = 90°? Wait, no, the right angle is at ∠5, so ∠5 is 90°, so ∠4 + ∠5 + ∠6? Wait, maybe I made a mistake. Wait, the problem: line \( m \parallel n \), line \( q \) perpendicular to \( p \), \( m\angle 1 = 40^\circ \), find \( m\angle 7 \). So ∠1 and ∠3 are vertical (40°), then ∠3 + ∠7 = 90° (because line \( q \) is perpendicular to \( p \), so the angle between \( q \) and \( p \) is 90°), so \( m\angle 7 = 90^\circ - 40^\circ = 50^\circ \)? Wait, no, wait, ∠1 and ∠3 are vertical (40°), then ∠3 and ∠7: since \( m \parallel n \), and \( q \) is a transversal, but \( q \) is perpendicular to \( p \), so the angle between \( q \) and \( m \) (or \( n \)): Wait, maybe ∠1 and ∠7: ∠1 is 40°, and ∠7 is 50° because 90° - 40° = 50°. Wait, yes, because line \( q \) is perpendicular to \( p \), so the angle between \( q \) and \( p \) is 90°, so ∠1 + ∠7 = 90°? Wait, ∠1 is 40°, so ∠7 = 90° - 40° = 50°? Wait, that makes sense. Because ∠1 and ∠7: since \( m \parallel n \), and \( q \) is a transversal, but \( q \) is perpendicular to \( p \), so the angle between \( q \) and \( m \) (∠1) and ∠7 (on \( n \)) add up to 90°? Wait, maybe. So step by step:

1. ∠1 and ∠3 are vertical angles, so \( m\angle 3 = m\angle 1 = 40^\circ \).
2. Line \( q \) is perpendicular to line \( p \), so \( \angle 3 + \angle 7 = 90^\circ \) (because they form a right angle with the perpendicular line).
3. Therefore, \( m\angle 7 = 90^\circ - 40^\circ = 50^\circ \).

Wait, that makes sense. So the key is that ∠3 (40°) and ∠7 add up to 90° because line \( q \) is perpendicular to \( p \), forming a right angle, so their sum is 90°.

Answer:

\( 50^\circ \)