Question

find the measure of \\( \angle p s r \\).
\\( m \angle p s r = \square ^ { \circ } \\)

Explanation:

Step1: Identify alternate interior angles

Since \( PQ \parallel NO \) (parallel lines with transversal \( ML \)), \( \angle PSR \) and \( \angle ORL \) are alternate interior angles? Wait, no, actually \( \angle PSR = (131 - 2x)^\circ \) and \( \angle ORL=(6x + 67)^\circ \)? Wait, no, looking at the diagram, \( PQ \) and \( NO \) are parallel, and \( ML \) is a transversal. Wait, actually, \( \angle PSR \) and \( \angle ORL \) – wait, no, maybe they are same - side interior angles? Wait, no, the angles \( (131 - 2x)^\circ \) and \( (6x + 67)^\circ \) should be equal? Wait, no, if the lines are parallel, alternate interior angles are equal. Wait, maybe I made a mistake. Wait, actually, \( PQ \) and \( NO \) are parallel, and \( ML \) is a transversal, so \( \angle PSR \) and \( \angle ORL \) – wait, no, let's check the direction of the angles. Wait, actually, the two angles \( (131 - 2x)^\circ \) and \( (6x + 67)^\circ \) are same - side interior angles? No, same - side interior angles are supplementary. Wait, that makes sense. Because if two parallel lines are cut by a transversal, same - side interior angles are supplementary. So \( (131 - 2x)+(6x + 67)=180 \).

Step2: Solve for x

Simplify the left - hand side: \( 131 - 2x+6x + 67 = 180 \)
Combine like terms: \( (131 + 67)+( - 2x+6x)=180 \)
\( 198 + 4x=180 \)
Subtract 198 from both sides: \( 4x=180 - 198 \)
\( 4x=- 18 \)? Wait, that can't be. Wait, maybe I got the angles wrong. Wait, maybe the angles are alternate interior angles, so they are equal. So \( 131 - 2x=6x + 67 \)
Subtract \( 6x \) from both sides: \( 131-2x - 6x=67 \)
\( 131 - 8x=67 \)
Subtract 131 from both sides: \( - 8x=67 - 131 \)
\( - 8x=- 64 \)
Divide both sides by - 8: \( x = 8 \)

Step3: Find \( m\angle PSR \)

Now substitute \( x = 8 \) into \( 131-2x \)
\( 131-2\times8=131 - 16 = 115 \)

Answer:

\( 115 \)