Parallelogram Proofs: Sides, Angles and Diagonals

One definition, several consequences

A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

That is the whole definition. It says nothing about side lengths, angle sizes or diagonals — and yet all of those turn out to be fixed by it. This lesson proves them.

Property	Proved from
opposite sides are congruent	a diagonal and ASA
opposite angles are congruent	the same congruence
consecutive angles are supplementary	same-side interior angles
diagonals bisect each other	a second pair of triangles and ASA

Throughout, $ABCD$ has vertices in order, so $\overline{AB} \parallel \overline{DC}$ and $\overline{AD} \parallel \overline{BC}$.

Opposite sides are congruent

Given: parallelogram $ABCD$. Prove: $AB = CD$ and $BC = DA$.

Draw diagonal $\overline{AC}$. It splits the parallelogram into $\triangle ABC$ and $\triangle CDA$.

Statement	Reason
$\overline{AB} \parallel \overline{DC}$ and $\overline{AD} \parallel \overline{BC}$	definition of parallelogram
$\angle BAC \cong \angle DCA$	alternate interior angles, $\overline{AB} \parallel \overline{DC}$
$AC = CA$	reflexive property
$\angle BCA \cong \angle DAC$	alternate interior angles, $\overline{AD} \parallel \overline{BC}$
$\triangle ABC \cong \triangle CDA$	ASA
$AB = CD$ and $BC = DA$	CPCTC

Each pair of parallel sides supplies one pair of equal angles, and the shared diagonal supplies the side between them. That is exactly ASA.

Opposite angles are congruent

The same congruence does most of this work.

$\angle B$ and $\angle D$ are corresponding parts of $\triangle ABC$ and $\triangle CDA$, so $\angle B \cong \angle D$ by CPCTC.

For the other pair, each of $\angle A$ and $\angle C$ is split by the diagonal into two pieces that were already shown equal:

$$m\angle A = m\angle BAC + m\angle DAC = m\angle DCA + m\angle BCA = m\angle C$$

Consecutive angles are supplementary

Angles next to each other, such as $\angle A$ and $\angle B$, sit on the same side of transversal $\overline{AB}$, between the parallel sides $\overline{AD}$ and $\overline{BC}$. Same-side interior angles of parallel lines add to $180°$.

$$m\angle A + m\angle B = 180°$$

So a parallelogram with one right angle has four: its neighbors are $180° - 90°$, and the angle opposite is equal to it.

The diagonals bisect each other

Given: parallelogram $ABCD$ with diagonals meeting at $E$. Prove: $AE = CE$ and $BE = DE$.

Statement	Reason
$AB = CD$	opposite sides of a parallelogram are congruent
$\angle BAE \cong \angle DCE$	alternate interior angles, $\overline{AB} \parallel \overline{DC}$
$\angle ABE \cong \angle CDE$	alternate interior angles, $\overline{AB} \parallel \overline{DC}$
$\triangle ABE \cong \triangle CDE$	ASA
$AE = CE$ and $BE = DE$	CPCTC

The first line uses the theorem proved two sections earlier. The order of the proofs matters: opposite sides had to come first, because this proof needs them.

In the figure above, $E$ is at $(4, 2)$. It is the midpoint of $\overline{AC}$, from $(0, 0)$ to $(8, 4)$, and of $\overline{BD}$, from $(6, 0)$ to $(2, 4)$.

Why equal diagonals mean a rectangle

A rectangle is a parallelogram with four right angles. Its diagonals are equal — and, more surprisingly, no other parallelogram’s are.

Rectangle → equal diagonals.

Statement	Reason
$AB = DC$	opposite sides of a parallelogram
$\angle ABC \cong \angle DCB$	all angles of a rectangle are right angles
$BC = CB$	reflexive property
$\triangle ABC \cong \triangle DCB$	SAS
$AC = DB$	CPCTC

Parallelogram with equal diagonals → rectangle.

Statement	Reason
$AC = DB$	given
$AB = DC$	opposite sides of a parallelogram
$BC = CB$	reflexive property
$\triangle ABC \cong \triangle DCB$	SSS
$\angle ABC \cong \angle DCB$	CPCTC
$m\angle ABC + m\angle DCB = 180°$	consecutive angles are supplementary
$\angle ABC$ is a right angle	congruent supplementary angles are right angles

One right angle in a parallelogram gives four, so $ABCD$ is a rectangle.

This is why a carpenter checks a frame by measuring its diagonals. A frame with equal opposite sides is a parallelogram; equal diagonals as well make it square at the corners — without ever measuring an angle.

Worked examples

Common mistakes

Practice problems