5. The given figure is an isosceles triangle $ABC (|AB|=|AC|)$, such that $|\angle BAC| > 50^{\circ}$. On the side $\overline{BC}$, a point $M$ is chosen such that $|\angle BAM| = 50^{\circ}$, and on the side $\overline{AC}$, a point $N$ is chosen such that $|AM| = |AN|$. What is the measure of the angle $\angle CMN$?
5. Sketch:

![](https://cdn.mathpix.com/cropped/2024_05_30_af61c3e47cfcd546148dg-2.jpg?height=528&width=580&top_left_y=1369&top_left_x=795)

$\angle C M A$ is the exterior angle of triangle $A M B$, so we have:

$$
x + y = \beta + 50^{\circ}
$$

$\angle A N M$ is the exterior angle of triangle $M C N$, so we have:
$y = x + \beta$
(2)
2 points

From (1) and (2), we have:

$x + x + \beta = \beta + 50^{\circ}$, 2 points

$2 x = 50^{\circ}$

$x = 25^{\circ}$,

The measure of angle $\angle C M N$ is $25^{\circ}$.