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# Parallelogram and its Properties

### Definition of Parallelogram

### Parallelogram Property 1

### Parallelogram Property 2

### Parallelogram Property 3

### Parallelogram Property 4

### Congruent lines

### Distance between two parallels

### Proving parallelogram method 1

### Proving parallelogram method 2

### Proving parallelogram method 3

### Proving parallelogram method 4

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.

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.

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.

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.

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.

The parallels between two parallels are congruent.

In the figure above, if l_{1} // l_{2} and AB // CD // EF, then AB = CD = EF.

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 l_{1} // l_{2} , P is a point lies on l_{1} , Q is a point lies on l_{2} and PQ is perpendicular to l_{2}, then PQ is the distance between parallel lines l_{1} and l_{2}.

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.

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.

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.

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.)