In the diagram, what is the value of $x?$ [asy]
draw((0,0)--(3,0)--(1,2.5)--cycle);
label("$60^\circ$",(2.9,0),NW);
label("$50^\circ$",(1.1,2.2),S);
label("$x^\circ$",(0,0),SW);
draw((-1,0)--(0,0));
draw((0,0)--(-.5,-1.25));
[/asy]
Since the angles in a triangle add to $180^\circ,$ then the missing angle in the triangle is $180^\circ-50^\circ-60^\circ=70^\circ.$ We then have: [asy]
draw((0,0)--(3,0)--(1,2.5)--cycle);
label("$60^\circ$",(2.9,0),NW);
label("$50^\circ$",(1.1,2.2),S);
label("$x^\circ$",(0,0),SW);
draw((-1,0)--(0,0));
draw((0,0)--(-.5,-1.25));
label("$A$",(-1,0),W);
label("$B$",(3,0),E);
label("$C$",(1,2.5),N);
label("$D$",(-.5,-1.25),S);
label("$X$",(0,0),NW);
[/asy] Since $\angle BXC=70^\circ,$ then $\angle AXC = 180^\circ - \angle BXC = 110^\circ.$

Since $\angle AXC = 110^\circ,$ then $\angle DXA = 180^\circ - \angle AXC = 70^\circ.$

Therefore, $x=\boxed{70}.$

(Alternatively, we could note that when two lines intersect, the vertically opposite angles are equal so $\angle DXA=\angle BXC =70^\circ.$)