In a quadrilateral WXYZ, ∠ W = 74° and ∠ X = 58°. The bisectors of ∠Y and ∠ Z meet at O. What is the measure of ∠ ZOY?

In quadrilateral WXYZ,

$\angle W = 74^\circ$
$\angle X = 58^\circ$

Sum of interior angles of a quadrilateral = $360^\circ$

So,

$\angle Y + \angle Z = 360^\circ - (74^\circ + 58^\circ)$

$\angle Y + \angle Z = 360^\circ - 132^\circ = 228^\circ$

The bisectors of $\angle Y$ and $\angle Z$ meet at O.

In any quadrilateral, the angle between the internal bisectors of two adjacent angles equals:

$\angle ZOY = 180^\circ - \frac{\angle Y + \angle Z}{2}$

Substitute the value:

$\angle ZOY = 180^\circ - \frac{228^\circ}{2}$

$\angle ZOY = 180^\circ - 114^\circ$

$\angle ZOY = 66^\circ$

Correct Answer: 66°