In triangle ABC, ∠A = 58° and ∠C = 62°. If O is the circumcentre, find ∠AOC.
120°
The sum of angles in triangle ABC is \(180^\circ\), so \(\angle B = 180^\circ - 58^\circ - 62^\circ = 60^\circ\).
Since O is the circumcentre, the arc AC subtends \(\angle B\) at the circumference and \(\angle AOC\) at the centre.
By the theorem that the angle subtended by an arc at the centre is twice the angle subtended at the circumference, \(\angle AOC = 2 \times \angle B\).
Substituting the value gives \(\angle AOC = 2 \times 60^\circ = 120^\circ\).
Hence, \(\angle AOC = 120^\circ\).
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