In circle $O$, $\overline{PN}$ and $\overline{GA}$ are diameters and m$\angle GOP=78^\circ$. How many degrees are in the measure of $\angle NGA$? [asy]
draw((0,1)..(1,0)..(0,-1)..(-1,0)..cycle);
draw((.777,.629)--(-.777,-.629));
draw((-.777,.629)--(.777,-.629));
draw((.777,.629)--(-.777,.629));
label("$O$",(0,0),S);
label("$N$",(.777,.629),NE);
label("$A$",(.777,-.629),SE);
label("$P$",(-.777,-.629),SW);
label("$G$",(-.777,.629),NW);
label("$78^\circ$",(0,0),2W);
[/asy]
Since $\overline{GA}$ and $\overline{PN}$ are diameters, point $O$ is the center of the circle.  We have $\angle AON = \angle GOP = 78^\circ$, so arc $AN$ has measure $78^\circ$.  Since $\angle NGA$ is inscribed in arc $AN$, we have $\angle NGA = \frac12\cdot 78^\circ = \boxed{39^\circ}$.