Question

two parallel lines are crossed by a transversal. what is the value of k? \\( \bigcirc \\ k = 9 \\) \\( \bigcirc \\ k = 20 \\) \\( \bigcirc \\ k = 60 \\) \\( \bigcirc \\ k = 71 \\) diagram: two parallel lines (y and z) with arrows, crossed by transversal x. angle at intersection with y: \\( (2k+11)^\circ \\), angle at intersection with z: \\( 131^\circ \\)

Explanation:

Step1: Identify angle relationship

Since lines \( y \) and \( z \) are parallel, and \( x \) is a transversal, the angle \( (2k + 11)^\circ \) and \( 131^\circ \) are corresponding angles (or alternate interior angles, depending on the diagram), so they are equal. Thus, we set up the equation:
\( 2k + 11 = 131 \)

Step2: Solve for \( k \)

Subtract 11 from both sides:
\( 2k = 131 - 11 \)
\( 2k = 120 \)

Divide both sides by 2:
\( k = \frac{120}{2} \)
\( k = 60 \)

Wait, no—wait, maybe I made a mistake. Wait, actually, if the angle with \( 131^\circ \) and \( (2k + 11)^\circ \) are same - side or corresponding? Wait, no, let's re - check. Wait, if the two parallel lines are cut by a transversal, and the angle \( (2k + 11)^\circ \) and \( 131^\circ \) are equal (corresponding angles), but wait, 2k+11 = 131? Wait, 2k=120, k = 60? But wait, let's check again. Wait, maybe the angle \( (2k + 11)^\circ \) and \( 131^\circ \) are equal. Wait, but let's do the calculation again.

Wait, the correct equation: If the two angles are equal (corresponding angles), then \( 2k + 11=131 \). Subtract 11: \( 2k = 120 \), divide by 2: \( k = 60 \). Wait, but the options have k = 60 as an option. Wait, but let me check again. Wait, maybe I misread the diagram. Wait, the angle with \( 131^\circ \) and the angle \( (2k + 11)^\circ \): if the lines are parallel, corresponding angles are equal. So yes, \( 2k+11 = 131 \), so \( 2k=120 \), \( k = 60 \).

Answer:

\( k = 60 \) (corresponding to the option \( k = 60 \))