Ambiguous Case

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Theorem

Ambiguous Case for Triangle Side-Side-Angle Congruence

Let $\triangle ABC$ be a triangle.

Let the sides $a, b, c$ of $\triangle ABC$ be opposite $A, B, C$ respectively.


Let the sides $a$ and $b$ be known.

Let the angle $\angle B$ also be known.

Then it may not be possible to know the value of $\angle A$.


This is known as the ambiguous case.


Ambiguous Case for Spherical Triangle

Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.

Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.


Let the sides $a$ and $b$ be known.

Let the angle $\sphericalangle B$ also be known.

Then it may not be possible to know the value of $\sphericalangle A$.


This is known as the ambiguous case (for the spherical triangle).