1. **State the problem:** We are given a diamond-shaped figure with points G, I, H, and J. We know that $m\angle GIH = m\angle GIJ = 61^\circ$ and the length $GJ = 46$. We need to find the length $GH$. 2. **Analyze the figure and given information:** The diamond shape suggests that $GI$ is vertical, and $GH$ and $GJ$ are horizontal segments from point G to points H and J respectively. The right angles at H and J imply $GH \perp HI$ and $GJ \perp IJ$. 3. **Use the given angles:** Since $m\angle GIH = m\angle GIJ = 61^\circ$, triangle $GIH$ and triangle $GIJ$ share the vertex I with equal angles at I. 4. **Consider triangle $GIJ$:** Since $GJ = 46$ and $m\angle GIJ = 61^\circ$, we can use trigonometric ratios to find $GI$. 5. **Use sine in triangle $GIJ$:** In triangle $GIJ$, $\sin(61^\circ) = \frac{GJ}{IJ}$. But we don't know $IJ$, so instead use cosine or sine with known sides. 6. **Use cosine in triangle $GIJ$:** $\cos(61^\circ) = \frac{GI}{IJ}$. We still need $IJ$. 7. **Use right angle at J:** Since $\angle J$ is right, triangle $GIJ$ is right-angled at J. So $GI$ is adjacent to angle $GIJ$, and $GJ$ is opposite. 8. **Apply tangent:** $\tan(61^\circ) = \frac{GJ}{GI} = \frac{46}{GI}$. Solve for $GI$: $$GI = \frac{46}{\tan(61^\circ)}$$ 9. **Calculate $GI$:** $\tan(61^\circ) \approx 1.804$, so $$GI \approx \frac{46}{1.804} \approx 25.5$$ 10. **Use triangle $GIH$:** It is similar to $GIJ$ with $m\angle GIH = 61^\circ$ and right angle at H. 11. **Apply tangent in $GIH$:** $\tan(61^\circ) = \frac{GH}{GI}$. Solve for $GH$: $$GH = GI \times \tan(61^\circ)$$ 12. **Calculate $GH$:** Using $GI \approx 25.5$ and $\tan(61^\circ) \approx 1.804$, $$GH \approx 25.5 \times 1.804 \approx 46$$ **Final answer:** $$GH = 46$$