Yes
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The word for the same in math is congruent. For example, if there are two squares, both the exact same size, length, width etc., then the two shapes are congruent.
Yes, if the perimeters are the same, then both of them have side = perimeter / 4. And of course the angles are all 90°, since they are squares.
The length is the length of the two congruent squares, 4. The width is the length of one square, 2. Area is length X width, or 4 X 2 = 8
I believe so, though I am not sure I can prove it.
Every parallelogram has. (That includes squares, rectangles, and rhombera.)