1. **Problem statement:** Given circle Q with \(\overline{MN} \cong \overline{OP}\) and radius 12 units, find the measure of \(\angle 4\). 2. **Relevant formula and rules:** In a circle, congruent chords subtend congruent arcs and angles. The radius is given as 12 units. Since \(\overline{MN} \cong \overline{OP}\), arcs \(\overset{\frown}{MN}\) and \(\overset{\frown}{OP}\) are congruent, so angles subtended by these arcs at the center or circumference are equal. 3. **Step-by-step solution:** - Since \(\overline{MN} \cong \overline{OP}\), \(m\angle 4 = m\angle 4\) (the angle subtended by chord \(OP\) at point 4). - The radius is 12, so the triangle formed by the radius and chord is isosceles. - Without additional numeric data for \(\angle 4\), and given the congruency, \(m\angle 4\) equals the measure of the angle subtended by \(\overline{MN}\) or \(\overline{OP}\). **Final answer:** \(m\angle 4 = m\angle 4\) (equal to the angle subtended by congruent chords, exact numeric value depends on further data).