# Parallelograms

Properties, Shapes, Diagonals and Area

A parallelogram is quadrilateral comprised of two pairs of parallel lines. There are several rules involving

### Interactive Parallelogram

The full size interactive parallelogram has additional features such as the lengths of the diagonals.

### Two Pairs of Parallel Lines

To create a parallelogram just think of 2 different pairs of parallel lines intersecting. ABCD is a parallelogram.

### Angles of A Parallelogram

#### Opposite Angles are Congruent

$$\angle D \cong \angle B \\ \angle A \cong \angle C$$

Triangles can be used to prove this rule about the opposite angle.

#### Consecutive angles are supplementary

The following pairs of angles are supplementary

$$\angle C$$ and $$\angle D$$
$$\angle C$$ and $$\angle B$$
$$\angle A$$ and $$\angle B$$
$$\angle A$$ and $$\angle D$$

To explore these rules governing the angles of a parallelogram use Math Warehouse's interactive parallelogram.

$$\angle A = 80°$$ since opposite angles are congruent.

You can then find $$\angle C$$ easily since it is opposite of $$\angle B$$

$$\angle C= 80°$$

### Sides of a Parallelogram

The opposite sides of a parallelogram are congruent

Triangles can be used to prove this rule about the opposite sides.

To explore these rules governing the sides of a parallelogram use Math Warehouse's interactive parallelogram

Since opposite sides are congruent you can set up the following equations:
2x − 10 = x + 80 and solve for x

x =90
3y − 4 = y + 20 and solve for y
y = 12

### Diagonals of a Parallelogram

The diagonals of a parallelogram bisect each other

AO = OD
CO = OB

To explore these rules governing the diagonals of a parallelogram use Math Warehouse's interactive parallelogram

Since the diagonals bisect each other, y = 16 and x = 22

x + 40 = 2x + 18
40 = x +18
40 = x + 18
22 = x

Classify Quadrilateral as parallelogram (a classic activity: have the students construct a quadrilateral and its midpoints, then create an inscribed quadrilateral. Then ask the students to measure the angles, sides etc.. of inscribed shape and use the measurements to classify the shape(parallelogram).

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