67. In triangle $ABC$, given are $\widehat{ABC}=100^{\circ}, \widehat{ACB}=$ $=65^{\circ}$. On $AB$, a point $M$ is taken such that $\widehat{MCB}=55^{\circ}$, and on $AC$ - a point $N$ such that $\widehat{NBC}=80^{\circ}$. Find $\widehat{NMC}$.
67. Describe a circle around $\triangle M C B$ (Fig. 13) and extend $B N$ until it intersects with the circle

![](https://cdn.mathpix.com/cropped/2024_05_21_6282cf4a54d2a9633c48g-080.jpg?height=310&width=450&top_left_y=486&top_left_x=634)

Fig. 12. at point $M_{1} ;\left|C M_{1}\right|=|C M|$, since the angles subtended by them $\left(80^{\circ}\right.$ and $\left.100^{\circ}\right)$ sum up to $180^{\circ}$; $\widehat{M_{1} C M}=\widehat{M_{1} B M}=20^{\circ}$, i.e., $N C$ is the bisector of angle $M_{1} C M$, and $\triangle M_{1} C N=\triangle N C M, \widehat{N M C}=\widehat{N M_{1} C}=\widehat{C M B}=25^{\circ}$.

Answer: $25^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_21_6282cf4a54d2a9633c48g-080.jpg?height=584&width=283&top_left_y=1024&top_left_x=270)

Fig. 13.

![](https://cdn.mathpix.com/cropped/2024_05_21_6282cf4a54d2a9633c48g-080.jpg?height=584&width=268&top_left_y=1024&top_left_x=742)

Fig. 14.