Calculate angles of an isosceles triangle knowing only the length of the base

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Is there a way to find the angles of an isosceles triangle if you only know the length of the base? Here's an image:

isosceles triangle

If I know the length in red, is it possible to figure out what the angles in purple are?

Probably not, so in that case, let's then say I know the lengths of the black lines if the first is impossible.