Unique SAS Triangle

Theorem: Given the lenghts of two sides and an included angle, a unique triangle can be formed.

Proof:
Let us draw a triangle $\triangle ABC$ with two sides AB and BC having lengths $l_1$ and $l_2$ respectively. Let the angle between them be $\theta$.

For proving this triangle to be unique for the given measurements, if possible, let us assume that there might be another triangle $\triangle A'BC$, such that BC $=l_1$, A'B $=l_2$ and angle $\angle A'BC = \theta$.

There can be three possibilities, viz, A' = A, A' lies on the line AB but not on the point A, or A' does not lie on the line AB.

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

If A' lies on the line AB but not on the point A (as shown below), then the length A'B $\neq$ AB. This is a contradiction as both A'B and AB equals $l_1$. Thus, this case is not possible.

If A' does not lie on the line AB (as shown below), then the angle $\angle A'BC \neq \angle ABC$. This is again a contradiction as both the angles are equal to $\theta$. Hence this case is also not possible.

Hence $\triangle ABC$ is unique.


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