To find the measure of angle \( \angle KIJ \), we can use the properties of arcs in a circle. The measure of an inscribed angle (an angle whose vertex is on the circumference of the circle) is half the measure of the intercepted arc.
In this case, arc \( KJ \) measures \( 116^\circ \). Since \( \angle KIJ \) intercepts arc \( KJ \), we can find \( m\angle KIJ \) by taking half of the arc measure:
\[ m\angle KIJ = \frac{1}{2} \times m\text{(arc KJ)} = \frac{1}{2} \times 116^\circ = 58^\circ \]
Therefore, the measure of \( \angle KIJ \) is \( 58^\circ \).
The correct response is:
58 degrees.