The problem asks to find the angle at which the angle bisectors of the acute angles in a right triangle intersect.

1. Problem Description

The problem asks to find the angle at which the angle bisectors of the acute angles in a right triangle intersect.

2. Solution Steps

Let the two acute angles of the right triangle be $\alpha$ and $\beta$.

Since it's a right triangle, we know that $\alpha + \beta = 90^{\circ}$.

The angle bisectors of these angles will be $\frac{\alpha}{2}$ and $\frac{\beta}{2}$.

Let the angle at which these bisectors intersect be $\gamma$.

Consider the triangle formed by the two angle bisectors and the line segment connecting the points where the bisectors meet the legs of the right triangle. The angles of this triangle are $\frac{\alpha}{2}$, $\frac{\beta}{2}$, and $\gamma$.

The sum of the angles in a triangle is $180^{\circ}$, so

$\frac{\alpha}{2} + \frac{\beta}{2} + \gamma = 180^{\circ}$.

We can rewrite this as

$\frac{\alpha + \beta}{2} + \gamma = 180^{\circ}$.

Since $\alpha + \beta = 90^{\circ}$, we have

$\frac{90^{\circ}}{2} + \gamma = 180^{\circ}$.

$45^{\circ} + \gamma = 180^{\circ}$.

$\gamma = 180^{\circ} - 45^{\circ}$.

$\gamma = 135^{\circ}$.

However, this is the obtuse angle. The acute angle is $180^{\circ} - 135^{\circ} = 45^{\circ}$. But that is incorrect because the intersection angle is clearly larger than a right angle. We are looking for the angle $\gamma$.

$\gamma = 135^{\circ}$.

3. Final Answer

135 degrees