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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 1

Rule 2: Opposite Sides are Congruent Read more Rule 2

Rule 3: Opposite angles are congruent Read more Rule 3

Rule 4: Adjacent anglesare supplementary Read more Rule 4

Rule 5: Diagonals bisect each other Read more Rule 5

Interactive Parallelogram

∠ A
∠ B
∠ C
∠ D
AB
BC
CD
DA
AO
BO
CO
DO
Drag Points To Start Demonstration

Two Pairs of Parallel Lines

Parallelograms Shape and Properties

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

Click on the button below to turn the pure parallel lines into a parallelogram.

Parallelograms Shape and Properties

Angles of Parallelogram

Consecutive angles are supplementary

consecutive angles of a parallelogram

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

$$\angle Y = 40 ^{\circ}$$. What is the measure of angles X,W, and Z in parallelogram WXYZ?

Picture Angles of Parallelogram

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 $$ \angle Z $$

Problem 2

What is the measure of x, y, z in parallelogram below?

Paralelogram Angles Diagram

Sides of A Parallelogram

Sides of 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 3

What is the length of side BD and side CD in parallelogram ABCD?

Sides of Parallelogram Diagram
Problem 4

What is x in the parallelogram on the left?

Parallelogram Sides Picture
Sides Practice Problem
Problem 5

What is the value of x and y in the parallelogram below?

Sides Practice Problem

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

Parallelogram Diagonals Bisect Each Other

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

What is x and Y?

Parallelogram Diagonals

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

Problem 7

What is x?

Diagonals of Parallelogram Diagram

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


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