Properties of a parallelogram

The properties of a parallelogram are listed below. We will use a parallelogram ABCD to show these properties.

Parallelogram ABCD

Property #1

Opposite sides of a parallelogram are congruent.

The length of AB is equal to the length of DC.
The length of BC is equal to the length of AD.

Property #2

Opposite angles of a parallelogram are congruent.

Angle A is equal to angle C
Angle B = angle D

Property #3

The diagonals of a parallelogram bisect each other.

Diagonal AC ( red line ) intersects and bisects diagonal BD ( red line ) at E.

Diagonals of a parallelogram

Property #4

Consecutive angles are supplementary or add up to 180 degrees.

Angle A + angle B = 180 degrees
Angle B + angle C = 180 degrees
Angle C + angle D = 180 degrees
Angle D + angle A = 180 degrees

Property #5

Each diagonal of a parallelogram turns the parallelogram into 2 congruent triangles.

Congruent triangles of a parallelogram

Triangle ABC is congruent or identical to triangle ADC.
Triangle BCD is congruent or identical to triangle BAD.

Using the properties of a parallelogram to solve math problems

Example  #1: Use the parallelogram below to find the length of segment BC and segment AD.

find x in a parallelogram

Since the opposite sides of a parallelogram are congruent, the length of segment BC is equal to the length of segment AD.

4x - 10 = 3x + 5.

Subtract 3x from each side

4x - 3x - 10 = 3x - 3x + 5

Simplify each side

x - 10 = 5

Add 10 to both sides of the equation.

x - 10 + 10 = 5 + 10

Simplify

x = 15

BC = AD = 4x - 10 = 4 times 15 - 10 = 60 - 10 = 50

Example  #2: Use the parallelogram below to find the length of segment AC and segment BD.

Diagonals of a parallelogram

Since the diagonals of a parallelogram bisect each other, we get the following results:

  • The length of segment AI is equal to the length of segment CI 
  • The length of segment BI is equal to the length of segment DI 

This leads to a system of linear equations to solve

2y - 4 = 4x  

y = x + 4

Substitute x + 4 for y in 2y - 4 = 4x

2(x + 4) - 4 = 4x

Distribute

2x + 8 - 4 = 4x

Simplify

2x + 4 = 4x

Subtract 2x from each side

2x - 2x + 4 = 4x - 2x

Simplify

4 = 2x

x = 4/2 = 2

y = x + 4 = 2 + 4 = 6

AC = AI + CI = 2y - 4 + 4x = 2×6 - 4 + 4×2 = 12 - 4 + 8 = 16 

BD = BI + DI = x + 4 + y = 2 + 4 + 6 = 12 

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