SAS Theorem

Specifying two sides and the Angle between them uniquely determines a Triangle. Let $b$ be the base length and $h$ be the height. Then the Area is

$$K = {\textstyle{1\over 2}}ch = {\textstyle{1\over 2}}ac\sin B.$$ (1)

The length of the third side is given by the Law of Cosines,

$$b^2=a^2+c^2-2ac\cos B,$$

so

$$b=\sqrt{a^2+c^2-2ac\cos B}.$$ (2)

Using the Law of Sines

$${a\over\sin A}={b\over\sin B}={c\over\sin C}$$ (3)

then gives the two other Angles as

$$A = \sin^{-1}\left({a\sin B\over\sqrt{a^2+c^2-2ac\cos B}}\right)$$ (4)

$$C = \sin^{-1}\left({c\sin B\over\sqrt{a^2+c^2-2ac\cos B}}.\right)$$ (5)

See also AAA Theorem, AAS Theorem, ASA Theorem, ASS Theorem, SSS Theorem, Triangle