Can a rhombus and square ever be congruent?

Answer 1

Very clever question! The answer is yes, they can. A square is, in fact, a rhombus. It's a special case of one. A rhombus is a quadrilateral with four equal sides. A square fits that description except it also has four equal angles. So, if you have two identical squares, you can actually say that you have a rhombus that is congruent to a square! I hope your teacher didn't mark you wrong if you answered "no." If, however, you have two figures, one a square and the other a rhombus with one pair of obtuse angles and one pair of acute angles, then the answer is no.
Rhombus:

Rhombus is a quadrilateral with four equal sides.

Square:

Square is a quadrilateral with four equal sides and angle between the adjacent sides are 90o exactly.

This implies that square is always a rhombus.

But a rhombus need not be a square always.

A rhombus can be a square if the angles between the adjacent sides are 90o exactly.

So, a rhombus and square can be congruent if their sides are equal and the angle between the adjacent sides of a rhombus are 90o exactly. Source: www.icoachmath.com