Parallelogram – Properties and Quadrilaterals
A 4 sided polygon bounded by 4 finite line segments to form a closed figure is referred to as a quadrilateral. A special case of a quadrilateral is a parallelogram, in which opposite sides are parallel and equal to each other.
Fig. 1 shows a parallelogram in which AB||CD and AD||BC. Also, AD = BCand AB = CD. The diagonals AC and BD intersect at O. If parallel sides AB and CD are taken into consideration, and then diagonals AC and BD or the sides AD and BC can be considered as transversal to them. All the properties of parallel lines and transversal can be applied to parallel sides of a parallelogram.
Let us have a quick look at the properties of parallel lines and transversal.
In fig. 2, the line l intersects the parallel lines a and b at points P and Q. The line l is the transversal here.
- If a transversal intersects two lines which are parallel, then the pair of corresponding angles is equal.
From Fig. 2:∠1 = ∠6, ∠4 = ∠8, ∠2 = ∠5 and ∠3 = ∠7
- If a transversal intersects two lines which are parallel, then the pair of alternate interior angles is equal.
From Fig. 2: ∠4=∠5 and ∠3 =∠6
- If a transversal intersects two lines which are parallel then the pair of interior angles on the same side of the transversal are supplementary.
∠3 + ∠5 = 180°and ∠4 + ∠6 = 180°
Properties of Parallelogram:
The four basic properties are:
- Opposite sides of a parallelogram are equal
- Opposite angles of parallelogram are equal
- The diagonals divide it into two congruent triangles
- Diagonals bisect each other
The proof is explained as follows:
Consider a parallelogram ABCD as shown in figure below,
|1.||Draw diagonal AC and BD||Construction|
|2.||∠1 = ∠3||AB||CD, AC is a transversal. Alternate angles are equal.|
|3.||∠5 = ∠7||AD||BC, BD is a transversal. Alternate angles are equal.|
|4.||∠2 = ∠4||AD||BC, AC is a transversal. Alternate angles are equal.|
|5.||∠6 = ∠8||AD||BC, BD is a transversal. Alternate angles are equal.|
|6.||AC = AC and BD = BD||Identity|
|7.||∆ABC ≅ CDA and ∆ADB ≅ CDB||ASA (Angle Side Angle) Property|
|8.||AB = CD and AD = BC||From statement (5)(CPCT)|
|9.||∠A = ∠C and ∠D = ∠B||From statement (5)(CPCT)|
Note: CPCT stands for congruent parts of congruent triangles.
Thus, from above statements it can be seen that:
- Opposite sides are equal i.e. AB = CD and AD = BC
- Opposite angles are equal i.e. ∠A = ∠C and ∠D = ∠B
- Diagonals AC and BD divide the parallelogram ABCD into congruent triangles i.e. ∆ABC ≅ CDA and ∆ADB ≅ CDB.
Proof: Diagonals bisect each other
Consider a parallelogram ABCD with diagonals AC and BD intersecting at O as shown in the figure given below.
In ∆AOB and ∆COD:
|1.||∠3 = ∠5||Alternate angles are equal|
|2.||∠1 = ∠2||Vertically Opposite Angles|
|3.||AB = CD||Opposite sides are equal|
|4.||∆AOB ≅ ∆COD||AAS(Angle Angle Side) Property|
|5.||OB = OD and OA = OC||From statement (4)(CPCT)|
Thus, in parallelogram diagonals bisect each other.
The converse of the above statement is also true which states that: If diagonals bisect each other in a quadrilateral, then it is a parallelogram.
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