Question

in the figure, $\angle rqs \cong \angle qlk$. what is the value of $x$? \bigcirc 36 \bigcirc 72 \bigcirc 108 \bigcirc 144

Explanation:

Step1: Identify angle relationship

Since \( \angle RQS \cong \angle QLK \), and \( \angle RQS = x^\circ \), \( \angle QLK=(x - 36)^\circ \), also \( \angle RQS \) and the adjacent angle on line \( SP \) are supplementary, and \( \angle QLK \) and its adjacent angle on line \( KN \) are supplementary. But also, notice that \( \angle RQS \) and the angle adjacent to \( \angle QLK \) (linear pair or supplementary) - wait, actually, since \( \angle RQS \) and \( \angle QLK \) are congruent, and also \( \angle RQS \) and the angle at \( L \) (let's say \( \angle QLN \)) are corresponding angles if lines are parallel, but here we can also see that \( \angle RQS \) and the angle \( (x - 36)^\circ \) along with the straight line: Wait, actually, \( \angle RQS \) and the angle adjacent to \( (x - 36)^\circ \) (the vertical angle or supplementary) - no, better: Since \( \angle RQS \) and \( \angle QLK \) are congruent, and also \( \angle RQS \) and the angle that is supplementary to \( (x - 36)^\circ \) (wait, no). Wait, actually, \( \angle RQS \) and \( (x - 36)^\circ \) are related such that \( x+(x - 36)=180 \)? Wait, no, wait: Wait, \( \angle RQS \) is \( x^\circ \), and \( \angle QLK=(x - 36)^\circ \), but also, \( \angle RQS \) and the angle at \( L \) (the one with \( (x - 36)^\circ \)): Wait, actually, since \( \angle RQS \cong \angle QLK \), so \( x=x - 36 \)? No, that can't be. Wait, maybe I made a mistake. Wait, looking at the diagram, \( SQ \) and \( KL \) are parallel? Wait, \( SQ \) and \( KN \) are parallel? Wait, the lines \( SP \) and \( KN \) are parallel, cut by transversal \( RM \). So \( \angle RQS \) is a corresponding angle to \( \angle QLN \), and \( \angle QLK \) is adjacent to \( \angle QLN \) (linear pair). Wait, no, \( \angle QLK \) and \( \angle QLN \) are supplementary (since they form a linear pair), so \( \angle QLN = 180-(x - 36)^\circ \). But \( \angle RQS \) (which is \( x^\circ \)) is equal to \( \angle QLN \) (corresponding angles) because \( SP \parallel KN \). So \( x=180-(x - 36) \). Ah, that makes sense. So:

Step2: Set up equation

\( x=180-(x - 36) \)

Step3: Solve the equation

\( x = 180 - x+36 \)

\( x+x=180 + 36 \)

\( 2x=216 \)

\( x = 108 \)? Wait, no, wait: Wait, no, wait, if \( \angle RQS \cong \angle QLK \), then \( x=x - 36 \) is wrong. Wait, maybe the angles are supplementary? Wait, no, the problem says \( \angle RQS \cong \angle QLK \), so \( x=x - 36 \) is impossible. Wait, maybe I misread the diagram. Wait, \( \angle RQS \) is \( x^\circ \), and \( \angle QLK \) is \( (x - 36)^\circ \), and also \( \angle RQS \) and \( \angle QLK \) are same - side interior angles? No, same - side interior angles are supplementary. Wait, if they are same - side interior angles, then \( x+(x - 36)=180 \). Let's try that:

\( x+(x - 36)=180 \)

\( 2x-36 = 180 \)

\( 2x=180 + 36=216 \)

\( x = 108 \)? Wait, but the options have 108 as an option. Wait, let's check:

If \( x = 108 \), then \( x-36=72 \), and \( 108 + 72=180 \), which makes sense because they are supplementary (same - side interior angles), and since \( \angle RQS \cong \angle QLK \)? Wait, no, if they are supplementary, they can't be congruent unless they are 90 each. Wait, maybe the problem is that \( \angle RQS \) and \( \angle QLK \) are congruent, and also they are supplementary? That would mean \( x=x \) and \( x + x=180 \), so \( x = 90 \), but that's not an option. Wait, I must have misinterpreted the congruence. Wait, the problem says \( \angle RQS \cong \angle QLK \), so \( x=(x - 36) \)? No, that's impossible. Wait, maybe the diagram shows that \( \angle RQS \) and \( \angle QLK \) are alternate interior angles? Wait, if lines \( SP \) and \( KN \) are parallel, then alternate interior angles are congruent. So \( \angle RQS \) and \( \angle QLN \) are alternate interior angles, but \( \angle QLK \) is supplementary to \( \angle QLN \). Wait, no, \( \angle QLK \) and \( \angle QLN \) are linear pair, so \( \angle QLK+\angle QLN = 180 \). If \( \angle RQS \cong \angle QLN \) (alternate interior angles), and \( \angle RQS \cong \angle QLK \), then \( \angle QLN \cong \angle QLK \), so \( \angle QLK+\angle QLN = 180 \) implies \( 2\angle QLK = 180 \), so \( \angle QLK = 90 \), but that's not matching. Wait, maybe the problem is that \( \angle RQS \) and \( \angle QLK \) are congruent, and also \( \angle RQS \) and \( (x - 36)^\circ \) are such that \( x-(x - 36)=36 \), but no. Wait, let's look at the options. The options are 36, 72, 108, 144. Let's test \( x = 108 \): Then \( x-36 = 72 \), and \( 108+72 = 180 \), which is supplementary. If \( \angle RQS \) and \( \angle QLK \) are supplementary and congruent, then each would be 90, but 108 and 72 are supplementary, not congruent. Wait, the problem says \( \angle RQS \cong \angle QLK \), so \( x=x - 36 \) is wrong. Wait, maybe a typo, or maybe I misread the congruence. Wait, maybe \( \angle RQS \cong \angle QLN \), and \( \angle QLN \) is supplementary to \( \angle QLK \). So \( \angle RQS \cong \angle QLN \), so \( x=180-(x - 36) \), which is \( x=180 - x+36 \), \( 2x=216 \), \( x = 108 \). Ah, that must be it. So even though the problem says \( \angle RQS \cong \angle QLK \), maybe it's a mistake, and it's \( \angle RQS \cong \angle QLN \), or maybe \( \angle QLK \) is supplementary to \( \angle RQS \)'s congruent angle. So solving \( x + (x - 36)=180 \) gives \( 2x=216 \), \( x = 108 \).

Answer:

108