Two rectangles have the same area and the same perimeter. Does this mean that they are congruent to each other?

Yes. If two rectangles have the same area (A) and the same perimeter (P), they must be congruent (meaning their lengths and widths are identical).

Congruent Dimensions:
For rectangles, the Area ( $A = l \times b$ ) and Perimeter [$P = 2(l + b)$] systems of equations have only one unique solution for $l$ and $b$ .

Unique Shape:
If both $l_1 \times b_1 = l_2 \times b_2$ and $2(l_1 + b_1) = 2(l_2 + b_2)$ , the rectangles must be identical.

Counter-Example Logic:
While $a = 4 \times 4$ square ( $A = 16$ and $P = 16$ ) and $a = 2 \times 6$ rectangle ( $A = 12$ and $P = 16$ ) have the same perimeter, their areas differ.

If the areas are also forced to be 16, only the $4 \times 4$ square works, forcing congruence.