Identify properties of parallelograms

A parallelogram is a quadrilateral with two pairs of opposite sides parallel; consequently, opposite sides have equal lengths, opposite angles have equal measures, consecutive angles are supplementary, and the diagonals bisect each other. These properties can be identified in diagrams and used to distinguish parallelograms from other quadrilaterals, without assuming that the diagonals are equal or perpendicular; formal proof and classification of special parallelograms are beyond this scope.

Detailed Explanation: 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.

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Is Shape a Parallelogram (from image)

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