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Thus it is proved that the opposite sides of a parallelogram are congruent. From the same proof it can be said that the diagonal of a parallelogram divides it into two congruent triangles.

Since D ACB @ D CAD
Ð ABC @ Ð CDA

By drawing a diagonal from D to B it can be shown that Ð DAB @ Ð BCD which means that in a parallelogram the opposite angles are congruent.

Another important feature of a parallelogram is given in the theorem below :

Theorem: The diagonals of a parallelogram bisect each other. Figure 3.16 shows a parallelogram PQRS, seg.PR and seg.QS are its two diagonals that intersect in O.

Figure 3.16

To prove that seg.PR & seg.QS bisect each other at O.

In D SOR and D QOP, Ð OSR @ Ð OQP and Ð ORS @ Ð OPQ (alternate angles ).

SR @ PQ opposite sides of a parallelogram.

\ D SOR @ D QOP ( ASA )

\ seg. SO @ seg.OQ i.e. O is the midpoint of SQ and seg.PO @ seg.OR i.e. O is the midpoint of PR. Hence it is proved that PR and SQ bisect each other at O.

Summary of the properties of a Parallelogram

1) Both the pairs of opposite sides of a parallelogram are parallel to each other.

2) The opposite sides of a parallelogram are congruent.

3) The opposite angles of a parallelogram are congruent.

4) The two triangles, formed by a diagonal of a parallelogram, are congruent.

5) The diagonals of a parallelogram bisect each other.

Index

3. 1 Definition
3. 2 Terminology
3. 3 Sum Of Interior Angles Of A Polygon
3. 4 Sum Of Exterior Angles Of A Polygon
3. 5 Trapezoids
3. 6 Parallelogram
3. 7 Square, Rectangle And Rhombus

Chapter 4

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