Problem 1369

1. We start by noting that the two triangles are congruent right triangles with angles $40^\circ$, $50^\circ$, and $90^\circ$.
2. The hypotenuses of these triangles intersect to form the obtuse angle $\alpha$.
3. Consider the quadrilateral formed by the union of the two triangles. This quadrilateral has four angles: two right angles (each $90^\circ$) and two angles formed by the intersection of the hypotenuses.
4. The sum of the interior angles of any quadrilateral is $360^\circ$.
5. We know three of the angles in the quadrilateral: two are $90^\circ$ (from the right angles of the triangles), and the other two are the angles formed by the hypotenuses.
6. Let the two angles formed by the hypotenuses be $\alpha$ and $\beta$. Since the triangles are congruent and symmetric, $\alpha$ and $\beta$ are equal.
7. Therefore, we can write the equation for the sum of the angles in the quadrilateral as:
$$90^\circ + 90^\circ + \alpha + \beta = 360^\circ$$
8. Since $\alpha = \beta$, we can substitute $\beta$ with $\alpha$:
$$90^\circ + 90^\circ + \alpha + \alpha = 360^\circ$$
9. Simplifying, we get:
$$180^\circ + 2\alpha = 360^\circ$$
10. Subtracting $180^\circ$ from both sides, we find:
$$2\alpha = 180^\circ$$
11. Dividing both sides by 2, we get:
$$\alpha = 90^\circ$$
12. However, this is incorrect because $\alpha$ is supposed to be the obtuse angle formed by the hypotenuses. We need to reconsider the angles formed by the hypotenuses.
13. The correct approach is to consider the exterior angle formed by extending one of the hypotenuses. The exterior angle is supplementary to the interior angle formed by the hypotenuses.
14. The interior angle formed by the hypotenuses is $180^\circ - 90^\circ = 90^\circ$.
15. Therefore, the obtuse angle $\alpha$ is:
$$180^\circ - 90^\circ = 90^\circ$$
16. This is still incorrect. The correct approach is to consider the angles formed by the hypotenuses directly.
17. The correct measure of the obtuse angle $\alpha$ is:
$$360^\circ - (50^\circ + 50^\circ + 90^\circ) = 360^\circ - 190^\circ = 170^\circ$$

The final answer is $\boxed{170^\circ}$.