What method is used to prove that a triangle is isosceles?

To establish that a triangle is isosceles, the most direct approach is to demonstrate that at least two of its sides are congruent. This aligns with the definition of an isosceles triangle, which is characterized by having two sides of equal length. By utilizing the Pythagorean theorem, one can confirm the relationship between the sides of a triangle, especially in the case of a right triangle. However, in any triangle, if two sides are found to be equal, this clinches that the triangle is isosceles.

While measuring angles might reveal properties of the triangle, it does not provide direct evidence regarding the length of the sides required to classify it as isosceles. Similarly, calculating the area, while informative about the triangle's size, does not address the congruence of the sides. Using an equilateral triangle as a reference doesn’t aid in proving whether a specific triangle is isosceles, as it introduces a different classification of triangle altogether. Thus, showing that two sides are equal through the Pythagorean theorem is the most appropriate method to confirm that a triangle is isosceles.