In a quadrilateral PQRS, ∠P = \((4x + 8)^\circ\), ∠Q = \((3x + 2)^\circ\), ∠R = \(75^\circ\) and ∠S = \((x - 5)^\circ\). The shortest angle of the quadrilateral is
30°
The sum of the interior angles of any quadrilateral is 360°.
So (4x + 8) + (3x + 2) + 75 + (x – 5) = 360, i.e. 8x + 80 = 360, which gives 8x = 280 and x = 35.
The angles are ∠P = 4(35) + 8 = 148°, ∠Q = 3(35) + 2 = 107°, ∠R = 75° and ∠S = 35 – 5 = 30°.
Therefore the shortest angle is ∠S = 30°.
Check: 148 + 107 + 75 + 30 = 360°, and 30° is smaller than each of 75°, 107° and 148°.
Hence, 30°.
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