How can one prove a triangle is a right triangle?

A triangle can be proven to be a right triangle by applying the Pythagorean theorem, which states that in a right 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. If the side lengths of a triangle fulfill this relationship, it conclusively proves that the triangle has a right angle.

In this context, simply showing that a triangle has equal angles does not necessarily confirm it is a right triangle, as equal angles can indicate an equilateral triangle or other configurations. Measuring the angles can tell you if one of the angles is exactly 90 degrees, but it does not provide a mathematical relationship relating the sides. Likewise, proving congruency of all three sides only confirms that the triangles are identical in size and shape, which does not specifically indicate any right angles. Thus, using the Pythagorean theorem is the most definitive and universally applicable method for proving that a triangle is a right triangle.