Parallelogram

In a parallelogram; opposite sides are equal, opposite angles are equal and diagonals bisect each other.

Properties of a Parallelogram

Prove that in a parallelogram: 

(i) the opposite sides are equal;
 

(ii) the opposite angles are equal;
 

(iii) diagonals bisect each other.
 

Proof:

Properties of a Parallelogram Diagram

Let PQRS be a parallelogram. Draw its diagonal PR.

In ∆ PQR and ∆ RSP,

∠1 = ∠4       (alternate angles) 

∠3 = ∠2       (alternate angles) 

and PR = RP  (common) 

Therefore, ∆ PQR ≅ ∆ RSP (by ASA congruence) 

⇒ PQ = RS, QR = SP and ∠Q = ∠S. 

Similarly, by drawing the diagonal QS, we can prove that 

∆ PQS ≅ ∆ RSQ

Therefore, ∠P = ∠R

Thus, PQ = RS, QR = SP, ∠Q = ∠S and ∠P = ∠R. 

This proves (i) and (ii) 

In order to prove (iii) consider parallelogram PQRS and draw its diagonals PR and QS, intersecting each other at O.

In OPQand ∆ ORS, we have

PQ = RS                            [Opposite sides of a parallelogram]

∠POQ = ∠ ROS                   [Vertically opposite angles]

∠OPQ = ∠ORS                    [Alternate angles]

Therefore, OPQ ∆ ORS   [By ASA property]

⇒ OP = OR and OQ = OS.

This shows that the diagonals of a parallelogram bisect each other.

The converse of the above result is also true, i.e.,

(i) If the opposite sides of a quadrilateral are equal then it is a parallelogram.

(i) If the opposite angles of a quadrilateral are equal then it is a parallelogram.

(i) If the diagonals of a quadrilateral bisect each other then it is a parallelogram.

Parallelogram

Parallelogram

Properties of a Rectangle Rhombus and Square

Problems on Parallelogram

Practice Test on Parallelogram


Parallelogram - Worksheet

Worksheet on Parallelogram



8th Grade Math Practice

From Parallelogram to HOME PAGE


Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.