\(ABCD\) is a parallelogram where \(P\) and \(Q\) are mid-points of opposite sides of \(AB\) and \(CD\). If \(AQ\) intersects \(DP\) at \(S\) and \(BQ\) intersects \(CP\) at \(R\), then prove that \(DPBQ\) is a parallelogram.
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S. No.
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Statement
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Reason
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1.
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\(PB \ || \ DQ\)
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Since
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2.
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\(P\) is the mid-point of \(AB\)
\(Q\) is the mid-point of \(CD\)
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3.
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\(AB = CD\)
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4.
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Half of the equal lines are equal.
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5.
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Using (2)
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6.
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\(DPBQ\) is a parallelogram
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Hence, proved.
Answer variants:
\(AB \ || \ CD\)
Opposite sides of a parallelogram are equal
A quadrilateral is a parallelogram,
if a pair of opposite sides is equal and parallel
if a pair of opposite sides is equal and parallel
\(AB = \frac{1}{2} AB\) and \(DQ = \frac{1}{2} CD\)
\(PB \ || \ DQ\)
\(PB = \frac{1}{2} AB\) and \(DQ = \frac{1}{2} CD\)
A quadrilateral is a parallelogram,
if a pair of opposite sides is equal and perpendicular
if a pair of opposite sides is equal and perpendicular
\(AD \ || \ CB\)
All the sides of a parallelogram are equal