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Conditions for a parallelogram

The converse of the above theorems are proved below. These theorems give the conditions under which a quadrilateral is a parallelogram.

Theorem: A quadrilateral is a parallelogram, if its opposite sides are congruent. Figure 3.17 shows a quadrilateral ABCD with its opposite sides congruent.

Figure 3.17

To prove that ABCD is a parallelogram, join A to C and consider D ADC & D CBA

AD @ CB and DC @ BA ( given ) also AC @ CA ( same side )

\ D ADC @ D CBA ( SSS ).

\ Ð ACB @ Ð CAD and Ð ACD @ Ð CAB ( corresponding angles of congruent triangles are congruent ).

Ð ACB @ Ð CAD Þ AD çç BC because they are alternate angles formed by the transversal CD that intersects BC and AD. Since they are congruent the two lines intersected by the transversal are parallel.

Similarly it can be show that since Ð ACD @ Ð CAB AB çç DC. Since both the opposite sides are parallel to each other ABCD is a parallelogram.

Theorem: A quadrilateral is a parallelogram if itís diagonals bisect each other. Figure 3.18 shows a quadrilateral PQRS such that its diagonals seg.PR and seg.QS bisect each other on the point of intersection O.

Figure 3.18


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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