Parallelograms · Little Blue Book, Visualized

Try this

A parallelogram is a quadrilateral whose two pairs of opposite sides are parallel.

Drag $B$ or $C$ . The figure shows the lengths of the four sides: $A B$ and $C D$ are always equal, and so are $B C$ and $D A$ .
Look at the diagonal readings: “Bisect each other” is always ✓, while “equal” and “perpendicular” come and go.
Click “Angles”: opposite angles are equal, and neighboring angles add up to $18 0^{\circ}$ .
Drag $C$ straight above $B$ , then adjust so that $B C = A B$ : more labels light up in the family tree. The special parallelograms are the topic of the next two stops.

Predict first

A quadrilateral has one pair of opposite sides parallel and the other pair equal. Must it be a parallelogram?

Yes, alwaysNot necessarilyNo, never

Choose an answer first; then you can see the result.

Why: one diagonal, two congruent triangles

The diagonals of a parallelogram bisect each other

Step 1 / 4

First, opposite sides are equal
Join $A C$ . $A B ∥ D C$ gives $∠ B A C = ∠ D C A$ , $B C ∥ A D$ gives $∠ B C A = ∠ D A C$ , and $A C$ is shared, so $△ A B C ≅ △ C D A$ (ASA).

$A B = C D, B C = D A$
Draw both diagonals
Angle, side, angle
Read off the conclusion

The other way round, each of the following is enough to prove that a quadrilateral is a parallelogram:

both pairs of opposite sides are parallel (the definition);
both pairs of opposite sides are equal;
one pair of opposite sides is parallel and equal;
both pairs of opposite angles are equal;
the diagonals bisect each other.

Remove one condition

Break it

Counterexample lab: remove one condition

Conclusion: $A B C D$ is a parallelogram

$D$ is locked at the position $A + C - B$ , which makes $D C$ parallel and equal to $A B$ . However you drag, it is a parallelogram.

Practice

Exercise 1Basic

In parallelogram $A B C D$ , $∠ A = 7 0^{\circ}$ . Find $∠ B$ in degrees.

Exercise 2Basic

Parallelogram $A B C D$ has perimeter $28$ , and $A B : B C = 3 : 4$ . Find the length of $A B$ .

Exercise 3Advanced

The diagonals of parallelogram $A B C D$ are $A C = 10$ and $B D = 16$ , meeting at $O$ , and $A B = 7$ . Find the perimeter of $△ A O B$ .

Exercise 4Challenge

In parallelogram $A B C D$ , $A B = 5$ , $A D = 8$ , and the bisector of $∠ B A D$ meets $B C$ at $E$ . Find the length of $E C$ .

Mistake Museum · Exhibits from this stop

Before you leave, look at the traps other people fell into here.See every exhibit in the museum →

Exhibit

“One pair of opposite sides parallel, the other pair equal”

A B ∥ C D

and

A D = B C

, so quadrilateral

A B C D

is a parallelogram.

What went wrong? Think first, then open

Diagnosis One pair of opposite sides is parallel and the other pair is equal. An isosceles trapezoid satisfies both conditions, and it is not a parallelogram.

The criterion needs the same pair of opposite sides to be “parallel and equal”:

A B ∥ C D

and

A B = C D

. Or use “both pairs of opposite sides parallel” or “both pairs of opposite sides equal”.

“Parallel and equal must be about the same pair of sides”

Exhibit

Thinking the diagonals of a parallelogram are equal

A B C D

is a parallelogram, so

A C = B D

What went wrong? Think first, then open

Diagnosis The diagonals of a parallelogram bisect each other, but they are usually not equal. Equal diagonals are a property of rectangles.

Parallelogram:

O A = O C

and

O B = O D

. If you want

A C = B D

, first prove it is a rectangle.

“Bisecting each other ≠ equal”