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# Parallelograms

## Properties, Shapes, and Diagonals

A parallelogram is a quadrilateral made from two pairs of intersecting parallel lines. There are several rules involving:

Rule 1: Opposite sides are parallel Read more

Rule 2: Opposite Sides are Congruent Read more

Rule 3: Opposite angles are congruent Read more

Rule 5: Diagonals bisect each other Read more

### Interactive Parallelogram

Drag Points To Start Demonstration

### 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 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.

##### Problem 1

There are many different ways to solve this question. You know that the opposite angles are congruent and the adjacent angles are supplementary.

$$\angle \red W = 40^{\circ}$$ since it is opposite $$\angle Y$$ and opposite angles are congruent.

Since consecutive angles are supplementary
$$m \angle Y + m \angle Z = 180 ^{\circ} \\ 40^{\circ} + m \angle Z = 180 ^{\circ} \\ m \angle Z = 180 ^{\circ} - 40^{\circ} \\ m \angle \red Z = 140 ^{\circ}$$

$$\angle \red X = 140^{\circ}$$ since it's opposite $$\angle Z$$

### 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.

##### Problem 5

Since opposite sides are congruent you can set up the following equations and solve for $$x$$: $\text{ Equation 1} \\ 2x − 10 = x + 80 \\ x - 10 = 80 \\ x = 90$

Since opposite sides are congruent you can set up the following equations and solve for $$y$$: $\text{ Equation 2} \\ 3y − 4 = y + 20 \\ 2y − 4 = 24 \\ 2y = 24 \\ y = 12$

### Diagonals

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.

##### Problem 6

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

##### Problem 7

$$x + 40 = 2x + 18 \\ 40 = x +18 \\ 40 = x + 18 \\ 22 = x$$