O is the center of the given circle. Then ∠x + ∠y equals(a) 2 ∠

1.	O is the center of the given circle. Then ∠x + ∠y equals(a) 2 ∠z (b) \(\frac{∠z }{2}\) (c) ∠z (d) None of these
Answer» Answer : (c) ∠z ∠QSR = ∠QTR = \(\frac{1}{2}\) ×∠QOR = \(\frac{z}{2}\) (Angle subtended by an arc of a circle at the centre is double the angle subtended by it at any point on the remaining part of the circle) ∠PSM = ∠PTM = 180º - \(\frac{z}{2}\) (Straight line) Also, ∠SMT = ∠QMR = y (vert. opp. ∠s) ∴ In quad. PSMT, ∠SMT + ∠PTM + ∠TPS + ∠PSM = 360° ⇒ y + 180° – \(\frac{z}{2}\) + x + 180° – \(\frac{z}{2}\) = 360° ⇒ x + y = z.

O is the center of the given circle. Then ∠x + ∠y equals(a) 2 ∠z (b) \(\frac{∠z }{2}\) (c) ∠z (d) None of these

Answer»

Answer : (c) ∠z

∠QSR = ∠QTR = \(\frac{1}{2}\) ×∠QOR = \(\frac{z}{2}\)

(Angle subtended by an arc of a circle at the centre is double the angle subtended by it at any point on the remaining part of the circle)

∠PSM = ∠PTM = 180º - \(\frac{z}{2}\) (Straight line)

Also, ∠SMT = ∠QMR = y (vert. opp. ∠s)

∴ In quad. PSMT, ∠SMT + ∠PTM + ∠TPS + ∠PSM = 360°

⇒ y + 180° – \(\frac{z}{2}\) + x + 180° – \(\frac{z}{2}\) = 360°

⇒ x + y = z.

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