1. **State the problem:** We have an isosceles triangle $\triangle GHI$ where sides $GH \cong IG$ and the measure of angle $\angle I = 51^\circ$. We need to find the measure of angle $\angle G$. 2. **Recall properties of isosceles triangles:** In an isosceles triangle, the angles opposite the equal sides are also equal. Since $GH \cong IG$, the angles opposite these sides, which are $\angle I$ and $\angle G$, must be equal. 3. **Set up the equation:** Let $m\angle G = x$. Since $m\angle I = 51^\circ$, and $m\angle G = m\angle I$, we have $x = 51^\circ$. 4. **Use the triangle angle sum property:** The sum of interior angles in any triangle is $180^\circ$. So, $$x + x + m\angle H = 180^\circ$$ which simplifies to $$2x + m\angle H = 180^\circ$$ 5. **Find $m\angle H$:** Substitute $x = 51^\circ$ into the equation: $$2(51^\circ) + m\angle H = 180^\circ$$ $$102^\circ + m\angle H = 180^\circ$$ 6. **Solve for $m\angle H$:** $$m\angle H = 180^\circ - 102^\circ = 78^\circ$$ 7. **Final answer:** The measure of angle $\angle G$ is $\boxed{51^\circ}$.