1. The problem states that we have a right-angled triangle with two equal-length sides. 2. In a right-angled triangle, one angle is always $90^\circ$. 3. Since the two sides are equal, the triangle is also isosceles, meaning the other two angles are equal. 4. Let each of the equal angles be $x$ degrees. 5. The sum of angles in any triangle is $180^\circ$, so we have: $$90 + x + x = 180$$ 6. Simplify the equation: $$90 + 2x = 180$$ 7. Subtract 90 from both sides: $$2x = 90$$ 8. Divide both sides by 2: $$x = 45$$ 9. Therefore, the three angles are $90^\circ$, $45^\circ$, and $45^\circ$.