# Properties of a parallelogram

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

### How to prove that the opposite angles of a parallelogram are congruent angles

Given parallelogram ABCD, we need to prove ∠A ≅ ∠C; ∠B ≅ ∠D 1. Given

ABCD is a parallelogram

2. Consecutive angles are supplementary angles in a parallelogram (property #4)

∠A + ∠B = 180 degrees

∠B + ∠C = 180 degrees

3. Transitive property in properties of equality

∠A + ∠B = ∠B + ∠C

4. Subtraction property in properties of equality

∠A + ∠B - ∠B = ∠B - ∠B + ∠C

5. ∠A  =  ∠C

Using similar reasoning, you can show that ∠B ≅ ∠D

Property #3

The diagonals of a parallelogram bisect each other.

Diagonal AC (red line) intersects and bisects diagonal BD (green line) at E. ### How to prove that the diagonals of a parallelogram bisect each other

Given parallelogram ABCD, we need to prove that segment AC and segment BD bisect each other at E. 1. Given

ABCD is a parallelogram

2. Definition of parallelogram

Segment AB || segment DC

3. Parallel lines cut by a transversal form congruent alternate interior angles

∠1 ≅ ∠4; ∠2 ≅ ∠3

4. Opposite sides of a parallelogram are congruent

Segment AB ≅ segment DC

5. Triangle ABE is congruent to triangle CDE by ASA (angle-side-angle)

6. Segment AE ≅ segment CE and segment BE ≅ segment DE by CPCTC (corresponding parts of congruent triangles are congruent)

7. Definition of bisector

Segment AC and segment BD bisect each other at E.

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. 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. 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. 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

Check also the lesson about parallelogram to learn some interesting stuff about the parallelogram such as:

• What is a parallelogram?
• Types of parallelogram such as rhombus and rectangles
• Perimeter of a parallelogram
• Area of a parallelogram

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