Special quadrilaterals: parallelograms

A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Almost every notebook page, door, brick, and rectangle on a screen has this shape , and every one of them obeys the same four nice rules.

Concept

Formal definition: in parallelogram $ABCD$, $AB \parallel DC$ and $AD \parallel BC$.

From this single condition four wonderful properties follow:

Property 1: Opposite sides are equal. $AB = DC$ and $AD = BC$.

Why? Draw diagonal $AC$. Then $\triangle ABC$ and $\triangle CDA$ are congruent (by ASA, using the alternate angles formed by the parallels), so corresponding sides are equal.

Property 2: Opposite angles are equal. $\angle A = \angle C$ and $\angle B = \angle D$.

Why? The same triangle congruence in Property 1 makes the corresponding angles equal.

Property 3: Adjacent angles are supplementary. $\angle A + \angle B = 180^\circ$ (and similarly for the other three pairs).

Why? Sides $AD$ and $BC$ are parallel; $AB$ is a transversal cutting them; co-interior angles add to $180^\circ$.

Property 4: Diagonals bisect each other. If $M$ is the intersection of the diagonals, then $AM = MC$ and $BM = MD$.

Why? Triangles $\triangle AMB$ and $\triangle CMD$ are congruent (alternate angles + opposite sides equal + vertically opposite angles), so $M$ is the midpoint of both diagonals.

Converse facts (any one of these is enough to make a quadrilateral a parallelogram):

• Both pairs of opposite sides equal.
• Both pairs of opposite angles equal.
• One pair of opposite sides equal and parallel.
• Diagonals bisect each other.

So if you measure two pairs of opposite sides and find them equal, you have a parallelogram without checking anything else.

Why parallelograms matter. Their stability and symmetry make them the building block of tilings, mechanical linkages (think of an extending ironing board), and the entire formalism of vectors. The area of a parallelogram is $\text{base} \times \text{height}$, the same formula that drives a huge chunk of geometry.

Worked examples

Example 1. In parallelogram $ABCD$, $\angle A = 65^\circ$. Find the other three angles.

• $\angle C = \angle A = 65^\circ$ (opposite angles).
• $\angle B = 180^\circ - 65^\circ = 115^\circ$ (adjacent).
• $\angle D = \angle B = 115^\circ$.

Example 2. In parallelogram $PQRS$, side $PQ = 8$ cm and $PS = 5$ cm. Find $QR$ and $SR$.

• Opposite sides equal: $QR = PS = 5$ cm and $SR = PQ = 8$ cm.

Example 3. The diagonals of a parallelogram meet at $O$. If $AO = 5$ cm and $BO = 4$ cm, find the full diagonal lengths $AC$ and $BD$.

• Diagonals bisect: $AC = 2 \cdot AO = 10$ cm and $BD = 2 \cdot BO = 8$ cm.

Example 4. In a parallelogram, one angle is twice the adjacent angle. Find both.

• Let the smaller be $x$. Adjacent are supplementary: $x + 2x = 180^\circ$.
• $3x = 180^\circ \Rightarrow x = 60^\circ$ and the other is $120^\circ$.

Try it yourself

1. In parallelogram $ABCD$, $\angle A = 110^\circ$. Find $\angle B, \angle C, \angle D$.
2. The sides of a parallelogram are $7$ cm and $5$ cm. What is its perimeter?
3. The diagonals of a parallelogram meet at $O$. If $AC = 14$ cm, find $AO$.
4. The angles of a parallelogram are in the ratio $2:3$. Find them.
5. Show that the diagonals of a parallelogram divide it into four triangles of equal area.
6. A quadrilateral has both pairs of opposite sides equal. Must it be a parallelogram?
7. In parallelogram $WXYZ$, $\angle W = 3x$ and $\angle X = 2x + 30^\circ$. Find $x$.
8. Sketch a parallelogram and label its diagonals. Mark all equal segments.

Activity / Insight

Linkage demo. Cut four cardboard strips: two of length $8$ cm and two of length $5$ cm. Use paper fasteners to pin their ends together into a closed figure (the $8$'s opposite, the $5$'s opposite). You have a parallelogram. Now wiggle it , it can flex into many shapes, all of them parallelograms, because opposite sides keep their lengths. This same property is what holds the changing shape of a folding ironing board.