In A Parallelogram, Opposite Sides Are Equal
In A Parallelogram, Opposite Sides Are Equal
Theorem 2: In parallelogram, opposite sides are equal.
GIVEN A parallelogram ABCD TO PROVE AB = CD and DA = BC Construction Join AC Proof Since ABCD is a parallelogram. Therefore, AB Now, AD
Again, AB
Noe, in <DAC = <BCA [From (i)] AC = AC [Common side] and, <DCA = <BAC [From (ii)] So, by ASA-criterion of congruence
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Converse Theorem: A quadrilateral is a parallelogram if its opposite sides are equal
Given : A quadrilateral ABCD in which AB= CD and AD= BC To Prove: ABCD is a Parallelogram. Const: Join AC Proof: In AC= AC [ Common ] AB = CD [Given] BC = AD [Given]
But they are alternate interior angles when AB and CD are straight lines and AC is the transversal. As they are equal therefore AB || CD and But they are alternate interior angles when BC and AD are straight lines and AC is the transversal. As they are equal therefore BC || AD. As both the opposite pair of sides are parallel therefore ABCD is a parallelogram. | |
Illustration: Given a triangle ABC lines PQ, QR and PR are drawn from vertex B, A and C such that PQ || AC, QR || BC and PR || AB. Prove that BC is half or QR.
Proof: BC || QR and PR || AB Therefore ABCR is a parallelogram as both the opposite pair of sides are parallel. BC = AR ------------(i) [ Opposite sides of a parallelogram are equal ] Similarly, BC || QR and PQ || AC Therefore AQBC is a parallelogram as both the opposite pair of sides are parallel. BC = AQ --------------(ii) [ Opposite sides of a parallelogram are equal ] From (i) and (ii) BC = AQ= AR QR = AQ+ AR = BC + BC = 2 BC |  |
In the following figure, which sides of the parallelogram are equal?
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| Right Option : D | |||
| View Explanation | |||
In the following figure, which sides are parallel?
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| Right Option : C | |||
| View Explanation | |||
In the following figure, if PS = QR, PQ = SR, then the quadrilateral PQRS is a ________________.
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| Right Option : C | |||
| View Explanation | |||
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