The given figure shows a circle with centre O. A, B, C, and D are points on the circle such that AB = CD. What is the measure of ∠OCD?

It is given that AB = CD.

It is known that equal chords of a circle subtend equal angles at the centre.

∴ ∠AOB = ∠COD

⇒ ∠COD = 84°

In ΔCOD, OC = OD (Radii of the same circle)

∴ ∠ODC = ∠OCD (In triangles, angles opposite to equal sides are equal)

By angle sum property of a triangle, we obtain

∠ODC + ∠OCD + ∠COD = 180°

⇒ 2∠OCD + 84° = 180°

⇒ 2∠OCD = 180° − 84° = 96°

$\therefore \angle \mathrm{OCD}=\frac{1}{2}\times 96°=48°$

Thus, the measure of ∠OCD is 48°.