back to Geometry

Parallelogram and its Properties

Definition of Parallelogram

A parallelogram is a quadrilateral having two pairs of parallel sides.

In the figure above, if AB // DC and AD // BC, then ABCD is a parallelogtam.

Parallelogram Property 1

Opposite angles of a parallelogram are congruent.

In the figure above, if ABCD is a parallelogram, then angle A = angle C, and angle B = angle D

Note: angle A and angle C are a pair of opposite angles. Angle B and angle D are a pair of opposite angles.

Parallelogram Property 2

Opposites sides of a parallelogram are congruent.

In the figure above, if ABCD is a parallelogram, than AB = DC and AD = BC

Note: AB and DC are a pair of opposite sides. AD and BC are a pair of opposides.

Parallelogram Property 3

The diagonals of a parallelogram bisect each other.

In the figure above, if ABCD is a parallelogram, AC and BD are its two diagonals, then AE = EC and BE = ED

Note: the two diagonal AC and BD intersect at point E.

Parallelogram Property 4

Consecutive pairs of angles of a parallelogram are supplementary.

In the figure above, if ABCD is a parallelogram then angle D + angle A = 180^o, angle A + angle B = 180^o.

Note: in parallelogram ABCD, angle D and angle A are a pair of consecutive angles and angle A = angle C, angle B = angle D.

Congruent lines

The parallels between two parallels are congruent.

In the figure above, if l₁ // l₂ and AB // CD // EF, then AB = CD = EF.

Distance between two parallels

Distance between two parallels is the line drawn from any point of a parallel to the distance of another parallel.

In the figure above, if l₁ // l₂ , P is a point lies on l₁ , Q is a point lies on l₂ and PQ is perpendicular to l₂, then PQ is the distance between parallel lines l₁ and l₂.

Proving parallelogram method 1

If both pairs of opposite sides of a quadrilateral are equal, then it is a parallelogram.

In the figure above, if AB = DC and AD = BC, then ABCD is a parallelogram.

Note: AB and DC are a pair of opposite sides. AD and BC are a pair of opposite sides.

Proving parallelogram method 2

If both pairs of opposite angles of a quadrilateral are equal, then it is a parallelogram.

In the figure above, if angle A = angle C and angle B = angle D, then ABCD is a parallelogram.

Note: angle A and angle C are a pair of opposite angles. Angle B and angle D are a pair of opposite angles.

Proving parallelogram method 3

If a pair of opposite sides of a quadrilateral are equal and parallel, then it is a parallelogram.

In the figure above, if AB = DC and AB // DC, then ABCD is a parallelogram.

Note: AB and DC are a pair of opposite sides.

Proving parallelogram method 4

If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

In the figure above, AC and BD are two diagonals which intersect at point E, if AE = EC and BE = ED, then ABCD is a parallelogram.

Example 1:

In the figure below, ABCD is a parallelogram. If angle A = 120^o, what is the degree measure of the angle D?

Solution

Since ABCD is a parallelogram (Given), so

angle A + angle D = 180^o

(Consecutive pairs of angles of a parallelogram are supplementary.)

angle D = 180^o - angle A

= 180^o - 120^o

= 60^o

Therefore, the degree measure of the angle D is 120^o

Example 2

In the figure below, ABCD is a parallelogram. DB is a diagonal. AE is perpendicular to DB at E and CF is perpendicular to DB at F. Prove AE = CF.

Proof

Since ABCD is a parallelogram (given), so AB = DC and AB // DC (a pair of opposite sides of a parallelogram are equal and parallel)

Since AB // DC, so angle ABD = angle CDB (alternate interior angles are congruent)

Since AE is perpendicular to BD and CF is perpendicular to BD (given), so angle AEB = angle CFD = 90^o (property of perpendicular lines)

Since point E lies on BD and point F lies on BD, so angle ABD = angle ABE, angle CDB = angle CDF

In triangle ABE and triangle CDF,

since angle AEB = angle CFD, angle ABE = angle CDF, AB = CD,

then triangle ABE is congruent to triangle CDF (AAS theorem)

so AE = CF (In two congruent triangles, congruent angles opposite congruent sides.)

Example 3

In the figure below, ABCD is a parallelogram. AC is a diagonal. If DE = BF, prove EG = FG.

Proof

Since ABCD is a parallelogram, so AB = DC and AB // DC (a pair of opposite sides of a parallelogram are equal and parallel)

Since BF = DE (given), so AB - BF = DC - DE, that is, AF = CE

Since AB // DC, so angle BAC = angle DCA and angle AFE = angle CEF (alternate interior angles are congruent)

Since point F lies on AB and point E lies on CD, AC and EF intersect at point G, so angle BAC = angle FAG and angle AFE = angle AFG. Same reason, angle CEF = angle CEG and angle ECA = angle ECG.

In triangle AFG and triangle CEG,

since angle FAG = angle EGC, AF = CE, angle AFG = angle CEG,

then triangle AFG is congruent to triangle CEG (ASA postulate)

so FG = EG (In two congruent triangles, congruent angles opposite congruent sides.)