The properties of a parallelogram are listed below. We will use a parallelogram ABCD to show these properties.
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.
Opposite angles of a parallelogram are congruent.
Angle A is equal to angle C
Angle B = angle D
The diagonals of a parallelogram bisect each other.
Diagonal AC ( red line ) intersects and bisects diagonal BD ( red line ) at E.
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
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.
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
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:
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
2x + 8 - 4 = 4x
2x + 4 = 4x
Subtract 2x from each side
2x - 2x + 4 = 4x - 2x
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