In an isosceles right-angled triangle, what is the relationship between the two sides that are perpendicular to each other?

In an isosceles right-angled triangle, one angle is a right angle (90 degrees), and the other two angles are equal, making it an isosceles triangle. In such a triangle, the two legs (sides) that are adjacent to the right angle are congruent (equal in length).

The right angle ensures that the triangle follows the Pythagorean theorem, where the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two legs (a² + b² = c²).

Since it's an isosceles triangle, two sides are equal, and since it's also a right-angled triangle, those equal sides are the legs.

Therefore, the two legs of an isosceles right-angled triangle are always equal in length.

Hence, the correct relationship between the two sides that are perpendicular to each other in an isosceles right-angled triangle is that they are always equal in length.