1. **State the problem:** We are given a circle with center O and two arcs: $m\angle JR = 65.1^\circ$ and $m\angle FL = 21.4^\circ$. We need to find the measure of the exterior angle $m\angle JHR$ formed outside the circle by lines $HJ$ and $HL$. 2. **Recall the exterior angle theorem for circles:** The measure of an exterior angle formed by two secants, two tangents, or a tangent and a secant intersecting outside a circle is half the difference of the measures of the intercepted arcs. 3. **Identify intercepted arcs:** The exterior angle $m\angle JHR$ intercepts arcs $JR$ and $GL$. Given $m\angle JR = 65.1^\circ$ and $m\angle GL = 21.4^\circ$. 4. **Apply the formula:** $$m\angle JHR = \frac{1}{2} |m\widehat{JR} - m\widehat{GL}|$$ 5. **Substitute values:** $$m\angle JHR = \frac{1}{2} |65.1^\circ - 21.4^\circ|$$ 6. **Calculate the difference:** $$m\angle JHR = \frac{1}{2} |43.7^\circ|$$ 7. **Simplify:** $$m\angle JHR = \frac{1}{2} \times 43.7^\circ = 21.85^\circ$$ 8. **Final answer:** The measure of the exterior angle $m\angle JHR$ is **21.85 degrees**.