Why is a rectangle a parallelogram, but a parallelogram is not a rectangle?

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?

• It's not the case that "a parallelogram is never considered a rectangle". Some parallelograms are rectangles, and some are not. But you cannot say in general that "every parallelogram is a rectangle", which is what would normally be meant by "an (arbitrary) parallelogram is a rectangle". – Nate Eldredge Jun 29 '14 at 16:50
• As they usually say, all rectangles are parallelograms but not all parallelograms are rectangles – ClassicStyle Jun 29 '14 at 16:51
• @Jordell Please understand that I don't have as great of a mathematical understanding as active members in this community. Even though if it sounds silly, I was not aware of how a parallelogram is actually a type of rectangle with special properties like a dog is a type of mammal with special properties. – user3758041 Jun 30 '14 at 14:13
• @user3758041 I think we exposed the key problem : the statement an X is never a Y and the other statement an X is not always a Y are logically different in English. – rschwieb Jun 30 '14 at 19:34
• Wow, 11 answer atm. just for a mid grade school question. A single one would have done it, IMO. – Mike Lischke Jul 1 '14 at 12:12

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.

• Brilliant answer, and great logic! – user3758041 Jun 29 '14 at 17:04
• @user3758041 it’s not brilliant, it’s common sense. – bolov Jun 30 '14 at 11:59
• @Bolov No need to be so rude. – user3758041 Jun 30 '14 at 17:21
• @user3758041 I am sorry, my intention wasn’t to be rude or offend you in any way, but merely a comment on the brilliance of the argument. – bolov Jun 30 '14 at 17:41
• It is common sense, yet it may be considered brilliant in its simplicity and correctness. – rubik Jun 30 '14 at 18:31

Why is a rectangle a parallelogram, but a parallelogram is not a rectangle ?

Why are all cats animals, but not all animals are cats ?

• This should be a comment, and even so please refrain from such rudeness. – user3758041 Jun 29 '14 at 18:04
• Not rude, just a very odd question. – Lucian Jun 29 '14 at 18:19
• Written text does not transmit tone. I edited it since people seemed to get the wrong idea. – Lucian Jun 29 '14 at 18:34
• I don't speak English as a first language either. :-) – Lucian Jun 30 '14 at 10:51
• I think it is perfectly valid to answer a question with a rhetorical question, which illustrates the answer by analogy, and this answer is a great example of that. Don't just look at the question mark at the end and shout 'comment!' – jwg Jul 1 '14 at 5:31

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.

It confused me that a parallelogram is never considered a rectangle, ...

This is simply not true. Some parallelograms are rectangles, in particular the ones that have ninety degree angles.

Rect- , from latin, means "right".
Rectangle = That has right angles.
And here you have a parallelogram without right angles: The same way that not all rectangles are squares, not all parallelograms are rectangles. A rectangle is a parallelogram with 4 right angles.

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.

A rectangle is a special case of a parallelogram (ie a rectangle is a parallelogram with angles of 90º). A rectangle HAS to have angles of 90º, but a parallelogram does not.

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.

protected by Zev ChonolesJul 28 '15 at 8:17

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