What is the minimum information needed to solve a 5 sided irregular polygon?

Question

For example, for a triangle all you need is an angle and the length of one side, and using that you can solve for the rest of the trianlge’s dimensions.

Answer 1

Let the polygon be $ABCDE$.

Seven elements are often sufficient. For example, it would suffice to know any 4 sides together with the 3 included angles, eg to know lengths $AB$, $BC$, $CD$ and $DE$ and angles $B$, $C$ and $D$.

Why? Because lengths $AB$ and $BC$ and $\angle B$ determine triangle $ABC$, and similarly for triangles $BCD$ and $CDE$. The triangles together fix the position of $E$ relative to $A$, so the whole polygon is determined.

On the other hand the following case shows that even eight elements could be insufficient. Suppose $\angle A = \angle B = \angle C = 90^o$, $\angle D = \angle E = 45^o$, $AB = 2$; $CD=1$; $DE = \sqrt{2}$. Then the lengths of $AE$ and $BC$, which are parallel, are indeterminate.