TITLE: Are there two non-congruent quadrilaterals with same sets of sides and angles?
QUESTION [7 upvotes]: Are there two non-congruent quadrilaterals with same sets of sides and angles? 

It is relatively easy to construct such pentagons. Trying to construct quadrilaterals allows for seemingly multiple degrees of freedom, yet my every attempt ends up not working out.

REPLY [3 votes]: Yes.  There are two degrees of freedom in the solution space, so we can add two extra constraints to make the computation easier.  Here’s a solution with a 90° angle and a 45° angle:

The side lengths, from longest to shortest, are
$$\begin{align*}
1&, \\
α &\approx 0.838045, \\
(2 + \sqrt2) + (6 + 4\sqrt2) α + \sqrt2 α^2 - (12 + 8\sqrt2) α^3 &\approx 0.454531, \\
(3 + 3\sqrt2) + (9 + 7\sqrt2) α - (2 + 2\sqrt2) α^2 - (16 + 12\sqrt2) α^3 &\approx 0.284498,
\end{align*}$$
where $α$ is the positive root of
$$1 - \sqrt2 - 2α - 2α^2 + (4 - \sqrt2)α^3 + 4α^4 = 0.$$