Parallelogram

The idea

A quadrilateral whose opposite sides are parallel is a parallelogram; a rectangle is a special parallelogram with all its angles 90°.

Only rectangles?

You know that the opposite sides of a rectangle are parallel. Now imagine four rails: two parallel rails one way, and two parallel rails another way. The enclosed four-sided shape has parallel opposite sides.

Can a quadrilateral have both pairs of opposite sides parallel and still not be a rectangle?

• No, then it must be a rectangle
• Yes, many such shapes exist
• Only if it is a square

Two pairs of rails

The orange rails are parallel. The blue rails are parallel too. Where they cross they make the quadrilateral ABCD. Drag B and D to move the rails.

A bigger family

A quadrilateral whose opposite sides are parallel is a parallelogram. A rectangle is a special parallelogram with all its angles 90°.

Two pairs of parallel rails always give a parallelogram, however you tilt them. The shape was a rectangle at one tilt only: when the rails crossed at 90°.

So every rectangle is a parallelogram, but most parallelograms are not rectangles.

Notes

A quadrilateral whose opposite sides are parallel is a parallelogram. A rectangle is a special parallelogram with all its angles 90°.

Check yourself

In quadrilateral ABCD, AB ∥ DC and AD ∥ BC, and ∠A = 70°. What is ABCD?

In quadrilateral PQRS, only PQ ∥ SR. The other two sides are not parallel. Is PQRS a parallelogram?

How many pairs of parallel sides does a parallelogram have?

Answer: 2

A parallelogram has 4 sides. Opposite sides are parallel, so there are 2 pairs of parallel sides.

Mira says: "A square is a parallelogram." Is she right?

• A parallelogram that is not a rectangle — correct. Yes! Both pairs of opposite sides are parallel, so it is a parallelogram. ∠A is not 90°, so it is not a rectangle.
• A rectangle. A rectangle needs all its angles to be 90°. Here ∠A = 70°.
• Not a parallelogram. A quadrilateral with both pairs of opposite sides parallel is a parallelogram.
• Yes, one pair of parallel sides is enough. A parallelogram needs both pairs of opposite sides to be parallel.
• No, only one pair of opposite sides is parallel — correct. Yes! A parallelogram needs both pairs of opposite sides to be parallel.
• Yes, because it is a quadrilateral. Every quadrilateral has four sides, but a parallelogram needs both pairs of opposite sides parallel.
• Yes, a square is a rectangle, so its opposite sides are parallel — correct. Yes! A square is a rectangle, and a rectangle is a parallelogram, so a square is a parallelogram.
• No, a square has equal sides and a parallelogram does not. Equal sides do not stop a shape from being a parallelogram. What counts is that opposite sides are parallel, and in a square they are.
• No, only shapes that are not rectangles are parallelograms. Rectangles are parallelograms too. They are the special ones with all angles 90°.