It confused me that a parallelogram is never considered a rectangle, yet a rectangle is considered a special case of a parallelogram.
How is this possible?
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I think you may be confused about necessity and sufficiency. E.g. every Irishman is a mammal, given that he meets the conditions to be a mammal: live young, etc. Yet, not every mammal is an Irishman. Take the delightful wallaby as an example.
In the same way, every rectangle is a parallelogram in that it satisfies the conditions to be such a figure: it is a quadrilateral with two pairs of parallel edges. Yet, not every parallelogram is a rectangle. For, just like the Irishman, a rectangle has stricter conditions for membership in its set: the rectangle must additionally have four right angles, and the Irishman must be from Ireland.
This figure from Wiki may help. Think of $S$ as the class of rectangles and $N$ as the class of parallelograms. Or equivalently, Irishmen and mammals.
A rectangle is considered a special case of a parallelogram because:
A parallelogram is a quadrilateral with 2 pairs of opposite, equal and parallel sides.
A rectangle is a quadrilateral with 2 pairs of opposite, equal and parallel sides BUT ALSO forms right angles between adjacent sides.
In a rectangle, it is imperative that each angle of the quadrilateral is 90°. This is not true for all parallelograms since isn't necessary that any of the angles is 90°. All things have special cases. By extension, you can say that a square is a special case of both a rectangle and a parallelogram: The condition for a parallelogram is only for opposite sides to be equal in length. You develop this further for a rectangle by making any and therefore all angles to be 90. Finally, for a square you impose that ALL the sides be equal, making it a special case of both! Try to work out the relation between a rhombus and the others, it should give you some more clarity.
Why is a woman a human being, but a human being is not a woman?
You must understand the exact meanings of the sentences about paralelograms and rectangles:
The statement is that every rectangle is a paralelogram, just like every woman is a human being. That means that some paralelograms (women) are rectangles (humans), but there can exist other paralelograms (humans) which are not rectangles (women). The statement tells you nothing about them.
From larger sets of objects to smaller, more specialized sets:
Quadrilaterals: closed polygons with 4 sides
Parallelograms: Quadrilaterals with opposite sides that are parallel
Rectangles: Parallelograms with right-angle corners
Squares: Rectangles with all sides of equal length
A square is a rectangle, but a given rectangle is not necessarily a square, etc. The squares are a subset of rectangles; the rectangles are a superset of squares. The same relationship holds for rectangles and parallelograms.