In a circle, chord AB subtends an angle of 50° at a point C on the circle. What is the angle subtended by the same chord AB at another point D on the same arc as C?

Correct — B. Angles subtended by one chord at points on the same arc are equal.

Let O be the centre, and take ∠AOB on the arc away from C and D.
∠AOB = 2 × ∠ACB = 2 × 50° = 100°
∠ADB = ½ × ∠AOB = ½ × 100° = 50° → option (b)

Why the others are wrong

• (a)
  55° — 55° would mean D sees chord AB differently from C. On the same arc both angles are half of one central angle, 100°, so they are equal.
• (c)
  45° — 45° would need a central angle of 90°, but ∠ACB = 50° already fixes it at 100°. Sliding the point along the arc does not change the angle.
• (d)
  40° — 40° is ∠OAB, the angle between the chord and a radius: (180° − 100°) ÷ 2 in the isosceles △OAB. The angle at D on the circle is 50°.

Angles in the same segment are equal. A chord cuts the circle into two segments, and every point on one arc sees the chord at the same angle.

The reason is the inscribed angle theorem: each such angle is half the central angle of the opposite arc, and that central angle does not move. Points on the other arc see the chord at 180° minus this angle.

Key facts

• Angles in the same segment of a circle are equal.
• Angle at the centre = 2 × angle at the circumference, for the same arc.
• Angles subtended by a chord at points on opposite arcs add up to 180°.

Common traps

• Thinking the angle changes as the point moves along the arc, when it stays 50° at every point of that arc.
• Placing D on the opposite arc and working out 130°, when the stem puts D on the same arc as C.

No directly related past PYQ was found.