An isosceles triangle is a triangle with two sides that are equal in length. This means that two angle will also be equal to each other. Is there any way that a triangle could have two equal angles, but not be an isosceles triangle?
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$\begingroup$Let $ABC$ be a triangle with $\angle BCA=\angle ABC$. Then observe that $ABC\equiv ACB$ (the order of the vertices is important), because they satisfy $ASA$ criterion. Therefore, we have that $CA=AB$, i.e. $ABC$ is isosceles.
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