Question

PQRS is a square whose diagonals PR and QS intersect at O.M is a point on QR such that OQ = MQ. Find the measures of ∠MOR and ∠QSR.

Answer 1

In ΔQOM,
∠OQM = 45°     ...(In square diagonals make 45° with the sides)
OQ = MQ
⇒ ∠QOM = ∠QMO (i) ...(equal sides have equal angles opposite to them)
∠QOM + ∠QMO + ∠OQM = 180°
∠QOM + ∠QOM + 45° = 180°
2∠QOM = 180° - 45°
∠QOM = 67.5°
In ΔQOR,
∠QOR = 90°     ...(diagonals bisect at right angles)
∠QOM + ∠MOR = 90°
67.5° + ∠MOR = 90°
∠MOR = 22.5°
In ΔROS,
∠OSR = 45°     ...(In square diagonals make 45° with the sides)
⇒ ∠QSR = 45°.