Why are squares rectangles but not all rectangles are squares?

Answer 1

A rectangle has four sides and four right-angled corners.

A square has four sides and four right-angled corners andhas all sides of equal length.
A rectangle is defined as having:

1. Four sides;
2. All angles equal at 90°;
3. Two pairs of opposite sides of equal length.

A square is defined as having:

1. Four sides;
2. All angles equal at 90°;
3. All four sides of equal length.

The first two properties are common to both the square and rectangle.

The third property differs:

For a square all sides are of equal length. They can be divided into two pairs of opposite sides of equal length, which is the third property of the rectangle. Thus all squares are rectangles.

For a rectangle, having two pairs of opposite sides of equal length means that the two pairs can have different lengths (as long as the sides in each pair are the same length). The two sets may have the same length, in which case the rectangle is a square, but if the two sets have different lengths then all four sides do not all have the same length, and so are not squares. Thus not all rectangles are squares.

Squares form a proper subset of the rectangles (squares ⊂ rectangles) where the two sets of pairs of opposite sides have equal length (ie all four sides are of equal length).