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To prove that PQRS is a parallelogram, consider D POS and D ROQ.

seg.PO @ seg.RO and seg.OS @ seg.OQ ( given )

Ð POS @ Ð ROQ ( vertical angles )

\D POS @ D ROQ

OPS @ Ð ORQ corresponding angles of congruent triangles are congruent.

These are alternate angles formed on seg.PS and seg.QR by the transversal seg.PR and since they are congruent PS çç QR. Similarly by showing that Ð OSR @ Ð OQP, it can be shown that PQ çç SR. Since both the pairs of opposite sides are parallel lines PQRS is a parallelogram.

Theorem: If one pair of opposite sides is parallel and congruent, the quadrilateral is a parallelogram.

In figure 3.19 there is a quadrilateral LMNO where seg.LM @ seg.NO and seg.LM@ seg.NO . To prove that LMNO is a parallelogram.

Figure 3.19

Since seg.LM çç seg.ON Ð LMO @ Ð NOM ( alternate angles )

In D LMO and D NOM

seg.LM @ seg.NO

Ð LMO @ Ð NOM

seg.MO @ seg.OM (Same side )

\ D LMO @ D NOM

\ Ð LOM @ Ð NMO ( corresponding angles of congruent triangles are congruent ).

But they are alternate angles formed by seg.MO on seg.MN and seg.LO and since they are congruent seg.LO çç seg.MN. Therefore LMNO is a parallelograme.

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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