Triangles and Trapeziums?

The required property is

$(\text{Slant Side})^2 = (\text{Opposite of Slant Side})^2+(\text{Difference of Parallel Sides})^2$

As we want them as integers, we have to find pythagorean triples $(a',b',c')$ with $a'<b'<c'$ , to assign $c'$ to Slant Side.

I will use the following fact,

Whenever $(a,b,c)$ with $a<b<c$ is a primitive pythagorean triple, $c$ is of the form $4n + 1$ . ... .. .. $(1)$

At least one Pythagorean Triple exists with $13$ in it as largest member; ( $(5,12,13)$ ).

At least one Pythagorean Triple exists with $15$ in it as largest member. (multiple of $(3,4,5)$ ).

At least one Pythagorean Triple exists with $17$ in it as largest member ( $(8,15,17)$ .

$\boxed{\text{No}}$ Pythagorean Triple exists with $\boxed{19}$ in it as largest member─ no primitive, because that would contradict fact- $(1)$ ─ no non-primitive, because $19$ is a prime.