Find an angle which is two-thirds of its complement.

Let the angle be x.

Its complement is (90 - x).

The problem states the angle is $\dfrac{2}{3}$ of its complement.

∴ x = $\dfrac{2}{3}(90^{\circ} - x)$

⇒ 3x = 2(90° - x)

⇒ 3x = 180° - 2x

⇒ 3x + 2x = 180°

⇒ 5x = 180°

⇒ x = $\dfrac{180^{\circ}}{5}$

⇒ x = 36°

Hence, the required angle measures 36°.