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In the diagram provided, \(PQRSTU\) is a regular hexagon, and \(SRMN\) and \(TUV\) are straight lines.

To find the value of \(x\), observe that:
- Since \(PQRSTU\) is a regular hexagon, each internal angle of the hexagon is \(120^\circ\).
- \(UV\) and \(SR\) are straight lines, hence \(\angle QPU\) and \(\angle QSR\) are supplementary (sum up to \(180^\circ\)).

Given \(\angle PQS = 80^\circ\) and \(\angle QSR = 135^\circ\), you can find \(\angle SQU\) by recognizing that \(\angle PQS + \angle SQU + \angle UQR = 180^\circ\) since \(SQU\) is a straight line.

We can calculate \(\angle SQU\) using the internal angle sum of triangle \(QSU\), where:
\[
\angle SQU = 180^\circ - (\angle PQS + \angle QSR) = 180^\circ - (80^\circ + 135^\circ) = -35^\circ
\]
Since angles on a straight line add up to \(180^\circ\), we actually need the exterior angle \( \angle UQR \), which is:
\[
\angle UQR = 180^\circ - 135^\circ = 45^\circ
\]

Therefore, \(\angle SQU\) (which is supplementary to \( \angle UQR \)) is:
\[
\angle SQU = 180^\circ - 45^\circ = 135^\circ
\]

This calculation seems incorrect because it leads to a contradiction in the straight line sum. Let's reassess:

If \(\angle QSR = 135^\circ\) and \(\angle PQS = 80^\circ\), then the missing angle in the triangle, \(\angle PSR\) (formed as \(\angle PQS\) + \(\angle QSR\) + \(\angle PSR = 180^\circ\)), calculates as:
\[
\angle

In Diagram 9, PQRSTU is a regular hexagon. SRMN and TUV are straight lines. Find the value of x.

This geometry problem involves finding the value of angle x in a regular hexagon PQRSTU with straight lines SRMN and TUV. Key concepts include angles in polygons, supplementary angles, and properties of straight lines. Suitable for Grade 9-10 students.

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