1. **Problem statement:** We have two chords WF and HG intersecting inside a circle at point E. Given angles are $\angle W = 55^\circ$ and $\angle G = 71^\circ$. We need to find the measure of the angle or arc indicated at point E. 2. **Relevant formula:** When two chords intersect inside a circle, the measure of the angle formed is half the sum of the measures of the arcs intercepted by the angle and its vertical angle. Mathematically, if two chords intersect at E, then: $$\angle E = \frac{1}{2} (\text{arc } WH + \text{arc } FG)$$ 3. **Explanation:** The angle formed by the intersecting chords is equal to half the sum of the measures of the arcs intercepted by the angle and its vertical angle. 4. **Given:** $\angle W = 55^\circ$, $\angle G = 71^\circ$. 5. **Find:** The measure of the angle at E, which is the angle formed by the intersection of chords WF and HG. 6. **Solution:** Since $\angle W$ and $\angle G$ are inscribed angles intercepting arcs, the arcs intercepted by these angles are twice their measures. So, $$\text{arc } WF = 2 \times 55^\circ = 110^\circ$$ $$\text{arc } HG = 2 \times 71^\circ = 142^\circ$$ 7. **Calculate angle at E:** $$\angle E = \frac{1}{2} (110^\circ + 142^\circ) = \frac{1}{2} (252^\circ) = 126^\circ$$ 8. **Answer:** The measure of the angle at point E is $126^\circ$.