What type of triangle is the Pythagorean theorem applicable to?

The Pythagorean theorem is specifically applicable to right triangles, which are defined as having one angle equal to 90 degrees. The theorem states that in such a triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This relationship holds exclusively for right triangles and is derived from the geometric properties unique to them.

For other types of triangles, such as acute triangles (where all angles are less than 90 degrees) and obtuse triangles (which have one angle greater than 90 degrees), the Pythagorean theorem does not apply in its original form. Although there are other relationships and laws, such as the Law of Cosines, that can be used to investigate those triangles, they do not satisfy the conditions laid out by the Pythagorean theorem. Equilateral triangles, which have all sides of equal length and all angles equal to 60 degrees, are also outside the applicability of this theorem.

Thus, the theorem's design and derivation cater solely to right triangles, making them the only valid case for its application.