Question

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

Explanation:

Step1: Identify Relationship

Since \( \angle RQS \cong \angle QLK \), lines \( SP \) and \( KN \) are parallel (corresponding angles). Also, \( \angle RQS = x^\circ \) and \( \angle QLK=(x - 36)^\circ \). But \( \angle RQS \) and the angle adjacent to \( \angle QLK \) (on the straight line) are supplementary? Wait, no—actually, \( \angle RQS \) and \( \angle QLN \) would be equal, but here, since \( \angle RQS \) and \( \angle QLK \) are equal, and \( \angle RQS \) and the angle at \( L \) (linear pair or supplementary? Wait, no—wait, \( \angle RQS \) and \( \angle QLK \) are equal, and \( \angle RQS \) and the angle \( (x - 36)^\circ \) are supplementary? Wait, no, let's re - examine.

Wait, \( \angle RQS = x^\circ \), and \( \angle QLK=(x - 36)^\circ \). But also, \( \angle RQS \) and the angle that is supplementary to \( \angle QLK \)? Wait, no—actually, since \( SP \parallel KN \), and \( RM \) is a transversal, \( \angle RQS \) and \( \angle QLN \) are equal (corresponding angles), but \( \angle QLK \) and \( \angle QLN \) are supplementary (linear pair). Wait, no, \( \angle QLK+(x - 36)^\circ = 180^\circ \)? No, wait, \( \angle QLK \) is \( (x - 36)^\circ \), and \( \angle RQS=x^\circ \), and since \( \angle RQS \) and \( \angle QLK \) are equal? No, wait the problem says \( \angle RQS\cong\angle QLK \), so \( x=(x - 36) \)? No, that can't be. Wait, maybe I made a mistake. Wait, \( \angle RQS \) and \( \angle QLM \) (vertical angles or something else). Wait, no, let's look at the diagram again.

Wait, \( \angle RQS = x^\circ \), and \( \angle QLK=(x - 36)^\circ \). Also, \( \angle RQS \) and the angle adjacent to \( \angle QLK \) (on the straight line \( KN \)): since \( SP \parallel KN \), \( \angle RQS \) and \( \angle QLN \) are equal (corresponding angles), and \( \angle QLK+\angle QLN = 180^\circ \) (linear pair). But \( \angle RQS=\angle QLN \) (corresponding angles), and \( \angle RQS = x \), \( \angle QLK=(x - 36) \). So \( x+(x - 36)=180 \)? Wait, no, \( \angle QLN = 180-(x - 36) \), and \( \angle RQS=x \), so \( x = 180-(x - 36) \)? Wait, no, the problem says \( \angle RQS\cong\angle QLK \), so \( x=(x - 36) \)? No, that's impossible. Wait, maybe I got the angles wrong.

Wait, \( \angle RQS \) is at \( Q \), between \( RQ \) and \( SQ \), and \( \angle QLK \) is at \( L \), between \( QL \) and \( KL \). Since \( SP \parallel KN \), and \( RQ \) and \( QL \) are parts of the same line (transversal), so \( \angle RQS \) and \( \angle QLK \) are equal (corresponding angles), but also, \( \angle RQS \) and the angle that is supplementary to \( \angle QLK \)? No, wait, let's set up the equation correctly. Since \( \angle RQS \) and \( \angle QLK \) are equal (given \( \angle RQS\cong\angle QLK \)) and also, \( \angle RQS \) and \( (x - 36)^\circ \) are supplementary? Wait, no, \( \angle RQS=x^\circ \), and \( \angle QLK=(x - 36)^\circ \), and since \( \angle RQS \) and \( \angle QLK \) are equal, that would mean \( x=x - 36 \), which is impossible. So I must have misidentified the angles.

Wait, maybe \( \angle RQS \) and the angle supplementary to \( \angle QLK \) are equal. Wait, \( \angle QLK=(x - 36)^\circ \), so the supplementary angle to \( \angle QLK \) is \( 180-(x - 36)=216 - x \). But \( \angle RQS=x^\circ \), and if \( \angle RQS \) is equal to that supplementary angle (because of parallel lines), then \( x = 216 - x \).

Step2: Solve for x

Solve the equation \( x=216 - x \).
Add \( x \) to both sides: \( x+x=216 - x+x \)
\( 2x = 216 \)
Divide both sides by 2: \( x=\frac{216}{2}=108 \)? Wait, no, that doesn't match. Wait, maybe another approach. Wait, the problem says \( \angle RQS\cong\angle QLK \), so \( x=(x - 36) \)? No, that's wrong. Wait, maybe the angles are supplementary. Wait, if \( SP \parallel KN \), then \( \angle RQS+\angle QLK = 180^\circ \) (consecutive interior angles). Since \( \angle RQS\cong\angle QLK \), then \( x+(x - 36)=180 \).

Step3: Solve the correct equation

Set up the equation: \( x+(x - 36)=180 \)
Simplify: \( 2x-36 = 180 \)
Add 36 to both sides: \( 2x=180 + 36=216 \)
Divide by 2: \( x = 108 \)? Wait, no, 216 divided by 2 is 108? Wait, 180+36=216, 216/2 = 108. But let's check. If \( x = 108 \), then \( \angle QLK=108 - 36 = 72^\circ \), and \( \angle RQS = 108^\circ \). Are they supplementary? 108+72=180, which makes sense for consecutive interior angles. So that works.

Wait, so the correct equation is \( x+(x - 36)=180 \) (since they are consecutive interior angles, supplementary, and equal? No, wait the problem says \( \angle RQS\cong\angle QLK \), so \( x=(x - 36) \) is wrong. Wait, no, I think I messed up the angle correspondence. Let's re - analyze the diagram.

Line \( SP \) and line \( KN \) are parallel (since \( \angle RQS\cong\angle QLK \), corresponding angles). Transversal \( RM \) intersects them at \( Q \) and \( L \). \( \angle RQS \) is at \( Q \), above \( SP \), and \( \angle QLK \) is at \( L \), below \( KN \). Wait, actually, \( \angle RQS \) and \( \angle QLM \) are vertical angles? No, \( \angle RQS \) and \( \angle PQL \) are vertical angles? Wait, no, \( \angle RQS=x^\circ \), and \( \angle QLK=(x - 36)^\circ \). Since \( SP \parallel KN \), \( \angle RQS \) and \( \angle QLN \) are equal (corresponding angles). \( \angle QLN \) and \( \angle QLK \) are supplementary (linear pair), so \( \angle RQS+(x - 36)^\circ = 180^\circ \). Since \( \angle RQS=x^\circ \), we have \( x+(x - 36)=180 \).

Solving \( 2x-36 = 180 \)
\( 2x=180 + 36=216 \)
\( x = 108 \)

Answer:

108