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Parallelogram and its Properties

Definition of Parallelogram

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

What is a parallelogram? How to find that a figure is a parallelogram?

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.

How to find a parallelogram? What are the consecutive pairs of angles of a parallelogram??

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.

How to find a parallelogram? What are the opposites sides of a parallelogram??

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.

How to find a parallelogram? What are the diagonals of a parallelogram?

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.

How to find a parallelogram? What are the consecutive pairs of angles of a parallelogram?

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

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.

How to identify that lines 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.

How to find the distance between two parallels lines?

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.

How to prove a parallelogram? How to find the equal opposites sides are congruent?

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.

How to prove a parallelogram? How to find that the consecutive pairs of angles are supplementary?

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.

How to prove a parallelogram? How to find that the angles are supplementary?

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.

How to prove a parallelogram? How to find that the diagonals are equal?

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 = 120o, what is the degree measure of the angle D?

How to prove a parallelogram? How to find that consecutive angles are suplementary?
Solution
Since ABCD is a parallelogram (Given), so
angle A + angle D = 180o
(Consecutive pairs of angles of a parallelogram are supplementary.)
angle D = 180o - angle A
= 180o - 120o
= 60o
Therefore, the degree measure of the angle D is 120o

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.

parallelogram example 2
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 = 90o (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.

parallelogram example 3

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.)