Finding the Angle in a Right Triangle with a Dividing Segment

From the midpoint of the hypotenuse of a right triangle, a perpendicular line is drawn to intersect one of the legs. Let the point of intersection be denoted as $P$. The segment connecting point $P$ to the endpoint of the other leg divides the angle at the vertex opposite the hypotenuse in the ratio of $2:5$, with the smaller angle adjacent to the hypotenuse. Determine the measure of this angle.