# 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 is not a answer, It is a question. It is rhetorical, I am aware. But it does not clearly illuminate to someone who doesn't understand. Remember many of our users (and future users), do not speak English as a first language. A good answer should be more clear. – Lyndon White Jun 30 '14 at 9:54
• 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
• @jwg I have concluded this should be a comment, because it is unclear and just too vague. – user3758041 Jul 1 '14 at 19:59
• @jwg: I too thought that this rhetorical question (about cats & animals) was enough to point up the difference, but the questioner’s comments show they did not get it. Perhaps this warns us to attend to what the question tells us about the level of the questioner’s understanding. – PJTraill Dec 19 '18 at 20:53

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.