In a circle, angle subtended by chord AB at the center is 80°. What is the measure of angle ∠ACB, where C lies on the circle?

Correct — A.
Rule: the angle at the centre is double the angle at the circumference, both standing on the same arc.

∠AOB = 80°
∠ACB = 80° ⁄ 2 = 40° → option (a).

Why the others are wrong

• (b)
  45° — 45° would need a central angle of 90°. The chord makes 80° at the centre, and half of that is 40°.
• (c)
  80° — 80° is the central angle itself. A point on the circle sees the chord at half that angle.
• (d)
  60° — 60° would need a central angle of 120°. It is not half of 80°, and a point on the minor arc sees 140°, not 60°.

A chord subtends at the centre double the angle it subtends at any point on the remaining part of the circle. So once the central angle is known, the angle at the circumference is half of it.

Every point on the major arc sees AB at the same 40°: angles in the same segment are equal.

Key facts

• Angle at the centre = 2 × angle at the circumference on the same arc.
• Angles in the same segment are equal.
• Opposite angles of a cyclic quadrilateral add to 180°.
• The angle in a semicircle is 90°.

Common traps

• Stopping at 80°, the central angle, without halving.
• Halving when the question gives the angle at the circumference and asks for the central one. The centre always carries the larger angle.

No directly related past PYQ was found.