A parallelogram is a quadrilateral whose opposite sides are parallel; that is the definition. Everything else about it can be proved with the congruent-triangle methods of Proofs with Congruent Triangles and the angle sums of Polygon Angle Sums.
Why it works
Draw diagonal $\overline{AC}$ in parallelogram $ABCD$. Because $\overline{AB} \parallel \overline{DC}$, the alternate interior angles $\angle BAC$ and $\angle DCA$ are congruent; because $\overline{AD} \parallel \overline{BC}$, so are $\angle DAC$ and $\angle BCA$. With the shared side $\overline{AC}$, $\triangle ABC \cong \triangle CDA$ by ASA. CPCTC then gives $AB = CD$, $BC = DA$ and $\angle B \cong \angle D$. Using the other diagonal the same way gives $\angle A \cong \angle C$. Finally, the triangles $AEB$ and $CED$ formed where the diagonals cross are congruent by ASA, so $AE = EC$ and $BE = ED$.
- Opposite sides are congruent.
- Opposite angles are congruent.
- Consecutive angles are supplementary (they are same-side interior angles between parallel sides).
- The diagonals bisect each other.
Key idea
In a parallelogram, opposite sides and opposite angles are congruent, consecutive angles add to $180^\circ$, and the diagonals cut each other in half.
Worked example: Opposite sides
$ABCD$ is a parallelogram with $AB = 3x + 4$ and $CD = 5x - 10$. Find $AB$.
- Opposite sides are equal: $3x + 4 = 5x - 10$, so $2x = 14$ and $x = 7$.
- $AB = 3(7) + 4 = 25$.
Answer: $AB = 25$
Worked example: Diagonals
The diagonals of parallelogram $PQRS$ meet at $T$. $PT = 2y + 3$ and $TR = 4y - 7$. Find $PR$.
- The diagonals bisect each other: $2y + 3 = 4y - 7$, so $y = 5$.
- $PT = 13$, so $PR = 2 \cdot 13 = 26$.
Answer: $PR = 26$
Worked example: Angles
In parallelogram $ABCD$, $m\angle A = (2x + 30)^\circ$ and $m\angle B = (3x + 10)^\circ$. Find all four angles.
- Consecutive angles are supplementary: $(2x + 30) + (3x + 10) = 180$, so $5x = 140$ and $x = 28$.
- $m\angle A = 86^\circ$ and $m\angle B = 94^\circ$. Opposite angles match: $m\angle C = 86^\circ$, $m\angle D = 94^\circ$.
Answer: $86^\circ$, $94^\circ$, $86^\circ$, $94^\circ$
Common mistake
Assuming the diagonals of every parallelogram are congruent or perpendicular. They always bisect each other, but in a long, slanted parallelogram one diagonal is much longer than the other, and they cross at an angle that is not $90^\circ$. Congruent diagonals happen only in rectangles, perpendicular ones only in rhombuses.