Question

note: figure is not drawn to scale. in the figure, line p is parallel to line q. what is the value of r?
○ 10°
○ 40°
○ 50°
○ 130°

Explanation:

Step1: Identify Alternate Interior Angles

Since line \( p \parallel q \) and a transversal cuts them, the \( 50^\circ \) angle and \( r^\circ \) are alternate interior angles? Wait, no, wait. Wait, actually, the angle adjacent to \( 50^\circ \) on line \( p \) and the angle related to \( r \). Wait, no, let's correct. The \( 50^\circ \) angle and the angle vertical to \( r \) or alternate? Wait, no, when two parallel lines are cut by a transversal, alternate interior angles are equal. Wait, the \( 50^\circ \) angle and \( r \) – wait, no, the angle that is supplementary? Wait, no, let's see. The \( 50^\circ \) angle and the angle adjacent to it on line \( p \) form a linear pair? No, wait, the transversal creates a \( 50^\circ \) angle with line \( p \), and since \( p \parallel q \), the corresponding angle or alternate interior. Wait, actually, the angle \( r \) is equal to \( 50^\circ \)? Wait, no, wait, maybe I made a mistake. Wait, no, the \( 50^\circ \) angle and \( r \) are alternate interior angles? Wait, let's visualize. Line \( p \) and \( q \) are parallel, transversal crosses them. The \( 50^\circ \) angle is above line \( p \), and \( r \) is below line \( q \) on the other side. Wait, no, actually, the angle \( r \) and the \( 50^\circ \) angle are alternate interior angles, so they should be equal? Wait, but the options have \( 50^\circ \) as an option. Wait, let's check again.

Wait, no, maybe the \( 50^\circ \) angle and the angle supplementary to \( r \)? No, wait, the vertical angle of \( r \) and the \( 50^\circ \) angle – wait, no, let's think again. When two parallel lines are cut by a transversal, alternate interior angles are congruent. So the \( 50^\circ \) angle and \( r \) are alternate interior angles, so \( r = 50^\circ \)? Wait, but let's confirm. The transversal makes a \( 50^\circ \) angle with line \( p \), so the alternate interior angle with line \( q \) should be equal, which is \( r \). So \( r = 50^\circ \).

Wait, but wait, maybe I messed up. Let's see the diagram. The \( 50^\circ \) is at the top, line \( p \), then the transversal goes down, intersects line \( q \), and \( r \) is on the left side of the transversal below line \( q \). So the angle above line \( p \) (50°) and the angle below line \( q \) (r°) – are they alternate interior? Yes, because alternate interior angles are between the two parallel lines, on opposite sides of the transversal. So yes, they should be equal. So \( r = 50^\circ \).

Step2: Confirm with Alternate Interior Angles Theorem

By the Alternate Interior Angles Theorem, when two parallel lines are cut by a transversal, alternate interior angles are congruent. So \( r = 50^\circ \).

Answer:

\( 50^\circ \) (Option: \( 50^\circ \))