If you intend 'dimensions' to mean units then whenever the two quantities are to be operated on each other then they must have the 'dimensions', refer to dimensional analysis
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Two lines intersect at one point. If in two dimensions, and they do not intersect they are parallel. The other option in two dimensions is they are the co-linear, that is they are the same line, in which case they intersect at all points.
yes
They must be the same length.
The area of a rectangle is equal to its length times its width. So any two rectangles for which these dimensions have the same product, the area is the same. For example, a rectangle that is two meters wide and three meters long and one that is one meter wide and six meters long will both have an area of six square meters.
Perimeters must be the same