Prove that two sides of the triangle are equal. This shows the triangle is isosceles and that the angles these sides make with the third side are equal. If one of these angles is proved as 45 degrees, the other must be 45 and the third is therefore 90 and the shape is an isosceles right triangle. A triangle's angles must add up to 180 degrees.
Prove that two angles on either end of a single side are equal. This can be done alternatively to prove that the sides themselves are equal. If the two angles are equal, the two lines are equal and the triangle is isosceles. Prove that one of these angles is equal to 45 degrees and that therefore the other is also the same and the third is a right angle. The shape is then an isosceles right triangle.
Prove that there is a right angle (90 degrees) in the triangle. The presence of a right angle in any triangle makes it a right triangle. If the two sides that create the right angle are equal, the other two angles are equal to 45 degrees and the shape is therefore an isosceles right triangle.
Show that the ratio among the smaller sides and the hypotenuse is 1:1:√2. This is a property of an isosceles right triangle.