Unique ASA Triangle

Theorem: Given the sizes of two angles and an included side, a unique triangle can be formed.

Proof:
Let there be a triangle $\triangle ABC$ such that AB $= l_1$, $\angle ABC = \alpha$ and $\angle BCA = \beta$. We prove that this triangle is unique.

If possible, let us assume that another triangle $\triangle A'BC$ can be formed such that A'B $= l_1$, $\angle A'BC = \alpha$ and $\angle BCA' = \beta$. Then the three cases arise, viz, A' $=$ A, A' lies on the line AB or AC, or A' doesn't lie on either of the lines AB or AC.

If A' $=$ A, then there is nothing to prove.

If A' lies on the line AB (figure I), then $\angle BCA' \neq\angle BCA$. Alternatively, if A' lies on the line AC, then $\angle A'BC \neq\angle ABC$. In both these cases, there is a contradiction, as from the assumption, $\angle A'BC = \angle ABC = \alpha$ and $\angle BCA' = \angle BCA = \beta$.

If A' doesn't lie on either of the two lines (figure II), then also there is a contradiction, as $\angle A'BC \neq\angle ABC$ and $\angle BCA' \neq\angle BCA$.

Therefore, $\triangle ABC$ is unique.


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