Suppose that a hyperbolic quadrilateral $ABCD$ satisfies

$h(A, B) = h(C, D), h(B, C) = h(A, D)$.

Mark each of the following claims about the quadrilateral as true or false:

1. Opposite angles of the quadrilateral are supplementary. (FALSE)

3. Opposite angles of the quadrilateral are equal (FALSE)

4. The diagonals of the quadrilateral bisect each other. (TRUE)
5. The diagonals of the quadrilateral are orthogonal. (TRUE)
6. Opposite sides of the quadrilateral cannot intersect (TRUE)

I'm not sure about the last one but if anyone can confirm that would be great.

(I'm assuming $h$ means hyperbolic distance).

You can certainly have a situation like

   A--B
\/
/\
C--D


where $|AB|=|CD|$ and $|BC|=|AD|$. Being in the hyperbolic plane doesn't prevent that.

For example you could construct it by drawing a circle with center $X$ and two diameters $AD$ and $BC$. Then triangles $AXB$ and $CXD$ are congruent (by SAS since the two angles at $X$ are vertical); in particular $|AB|=|CD|$.

Claims 4 and 5 are also false in this configuration.