In the figure, three lines intersect at point P. The angles shown are $A^{\circ }$, $B^{\circ }$, and $C^{\circ }$. The angle vertical to $A^{\circ }$ is $4k-10$. The angle $B^{\circ }$ is $2k+10$. The angle $C^{\circ }$ is $k+5$. If $A^{\circ }$, $B^{\circ }$, and $C^{\circ }$ are adjacent angles that form a straight line, what is the value of $k$?