1. **Problem statement:** Given that $m\angle LGH = 45^\circ$ and $m\angle GHJ = 45^\circ$, find $m\angle GMH$. 2. **Understanding the problem:** The points $L, G, H, J, M$ lie on a circle, and angles are formed by chords intersecting on the circle. 3. **Key formula:** The measure of an inscribed angle is half the measure of its intercepted arc. 4. **Step to find $m\angle GMH$:** - $\angle GMH$ intercepts arc $GH$. - Since $m\angle GHJ = 45^\circ$, arc $GH$ corresponds to $2 \times 45^\circ = 90^\circ$. - Therefore, $m\angle GMH = \frac{1}{2} \times 90^\circ = 45^\circ$. **Final answer:** $m\angle GMH = 45^\circ$.