Question

Does a rhombus have opposite angles that are supplementary?

Answer 1

Generally speaking, NO. These angles are equal.
But see the details below.

Explanation:

Let's consider a rhombus `ABCD` with `AB=BC=CD=DA`.
Draw diagonal `AC` and consider two triangles this diagonal breaks the rhombus into: `Delta ABC` and `Delta ADC`.
Obviously, these triangles are congruent by three sides.
Therefore, angles `/_ABC` and `/_ADC` are equal in measure.

Supplemental are angles that sum up to `180^o`. Equal in measure angles can be supplemental only in one case - when they both are `90^o`. Therefore, if our rhombus is a square, opposite angles are supplemental. Otherwise, they are not supplemental because they are equal in measure and not `90^o` each.