isoceles triangle theorem proof?

Question

any triangle is congruent to itself. Explain how this observation can be used to prove the isoceles triangle theorem.

if anyone can help that would be great. I really cant figure this out.

This is another question i had problems with so if u can help with this id appreciate it.

Prove that the square root of 13 is irrational. I know that 3n^2=m^2 but how would this prove its irrational??

the isosceles triangle theorem : in an isosceles triangle, the angles opposite the equal sides are equal.

Answer 1

Given that 2 sides of an isoceles triangle are congruent (the definition), start by bisecting the angle made by the two congruent sides and extending the bisector until is hits the third side.
The two triangles thus formed are congruent by SAS - "side" are the given congruent sides, the "angle" are the angles formed by bisecting the angle, and "side" is the bisector that is shared by the two triangles formed.
Now that we know that the two "inner" triangles are congruent, by CPCTC (corresponding parts of congruent triangles are congruent) the two "base" angles are congruent.
QED
Quick proof that sqrt(13) is irrational: Assume it is rational so that a^2 / b^2 = 13 and you'll get a contradiction pretty quickly. By RAA (reductio ad absurdum) you have the result.

Answer 2

What is the Isoceles Triangle Theorm?