1. **Problem Statement:** We are given a circle with points H, I, F, and G on its circumference. The arcs IF and HG measure 84° and 103° respectively. We need to find the measure of angle $\angle G$ inside the circle. 2. **Relevant Formula:** The measure of an inscribed angle in a circle is half the measure of its intercepted arc. That is, if an angle intercepts an arc of measure $x$, then the angle measure is $\frac{x}{2}$. 3. **Identify the intercepted arc for $\angle G$:** Angle $\angle G$ is formed by points F and H on the circle, so it intercepts the arc FH. 4. **Calculate the measure of arc FH:** The total circle is 360°. The arcs IF and HG are given as 84° and 103°. 5. **Sum of arcs IF and HG:** $$84^\circ + 103^\circ = 187^\circ$$ 6. **Remaining arcs FH and GI sum:** $$360^\circ - 187^\circ = 173^\circ$$ 7. **Since the quadrilateral is formed by points H, I, F, G, the arcs FH and GI are the remaining arcs. We assume $\angle G$ intercepts arc FH, so we take arc FH as 173° minus arc GI. But without arc GI, we consider $\angle G$ intercepts arc HF which is 103° (given). So $\angle G$ intercepts arc HF = 103°. 8. **Calculate $m\angle G$:** Using the inscribed angle theorem: $$m\angle G = \frac{103^\circ}{2} = 51.5^\circ$$ **Final answer:** $m\angle G = 51.5^\circ$