In the given figure, TU || PS and points Q and R lie on PS. Also, \(\angle PQT = x^\circ\), \(\angle RQT = (x - 50)^\circ\) and \(\angle TUR = (x + 25)^\circ\). What is the measure of \(\angle URS\)?
\(140^\circ\)
Concept — angles on a straight line and parallel lines cut by a transversal. Since Q and R lie on PS, the ray QP and ray QR point in opposite directions along a straight line, so any point T off the line makes \(\angle PQT + \angle RQT = 180^\circ\) (linear pair).
Step 1 — From the linear pair: \(x + (x - 50) = 180 \Rightarrow 2x = 230 \Rightarrow x = 115\).
Step 2 — Then \(\angle TUR = x + 25 = 115 + 25 = 140^\circ\).
Step 3 — TU is parallel to PS and UR is a transversal. The angles \(\angle TUR\) and \(\angle URS\) are alternate interior angles, so they are equal.
Step 4 — Therefore \(\angle URS = 140^\circ\).
Hence, \(140^\circ\).
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