Skill: Identify properties of parallelograms

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

Because of this, these properties are always true:

• Opposite sides have equal lengths.
• Opposite angles have equal measures.
• Consecutive angles add to $180^\circ$.
• The diagonals bisect each other, meaning they cut each other into equal halves.

Do not assume that the diagonals are equal in length or perpendicular.

Worked example

$ABCD$ is a parallelogram. The information shown is:

• $AB=8$
• $BC=5$
• $\angle A=65^\circ$
• The diagonals intersect at $E$
• $AC=14$ and $BD=10$

Find $CD$, $AD$, $\angle B$, $\angle C$, $\angle D$, $AE$, $CE$, $BE$, and $DE$.

Step 1: Find the opposite side lengths

Opposite sides of a parallelogram are equal.

So,

$$CD=AB=8$$

and

$$AD=BC=5$$

Step 2: Find the angles

Opposite angles are equal, so

$$\angle C=\angle A=65^\circ$$

Consecutive angles are supplementary, so their measures add to $180^\circ$:

$$\angle B=180^\circ-65^\circ=115^\circ$$

Angle $D$ is opposite angle $B$, so

$$\angle D=115^\circ$$

Step 3: Find the diagonal halves

The diagonals of a parallelogram bisect each other. Therefore, $E$ is the midpoint of both diagonals.

Since $AC=14$,

$$AE=CE=\frac{14}{2}=7$$

Since $BD=10$,

$$BE=DE=\frac{10}{2}=5$$

Therefore, the answers are:

$$CD=8,\quad AD=5$$ $$\angle B=115^\circ,\quad \angle C=65^\circ,\quad \angle D=115^\circ$$ $$AE=CE=7,\quad BE=DE=5$$

The important clues are the words opposite, consecutive, and bisect. Use the matching parallelogram property for each one.