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In school, I was always taught that given 3 pieces information with at least the length of one side, it was possible to determine all the measures of a triangle.

However, I recently stumbled on the following problem, to which I found two solutions:

Find all the measures of the triangle that has a $64\unicode{xB0}$ angle opposed to a $48\ cm$ side and adjacent to a $50\ cm$ side.

First solution -> The angles measure $64\unicode{xB0}$, $46.57\unicode{xB0}$ and $69.43\unicode{xB0}$

Second solution -> The angles measure $64\unicode{xB0}$, $5.43\unicode{xB0}$ and $110.57\unicode{xB0}$

So, what's the real minimum number of pieces of information to solve a triangle?

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2 Answers

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Knowing three sides, two angles and a side, or two sides adjacent to a given angle always give enough information to determine a triangle uniquely, by the SSS, ASA (and AAS), and SAS laws, respectively. Knowing two sides and an angle not between those two sides, or SSA, is a special case in the sense that there may be either $1$,$2$, or no triangles at all satisfying a given choice of lengths for two sides and an angle. In particular there will exist a unique triangle satisfying a SSA choice if the given side opposite to the angle is longer than the side adjacent to the angle.

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Solution of triangles Wikipedia

The case you have is SSA which is not uniquely specified.

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