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Properties of a Parallelogram - Property: The diagonals of a parallelogram bisect each other. (at the point of their intersection)

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The diagonals of a parallelogram bisect each other.

Given: ABCD is a parallelogram.

To Prove: AO = OC and BO = OD.

Construction: Draw diagonal AC and diagonal BD.

Proof:

ABCD is a parallelogram.
∴ AB || DC and AD || BC

Now,
AB || DC, AC is the transversal intersecting them at A and C respectively. 
∠BAC = ∠DCA                       ........(Alternate interior angles)
Thus, ∠BAO = ∠ DCO            .........(1)

As, AB || DC and BD is the transversal intersecting them at B and D respectively. 
∠ABD = ∠CDB                    ........(Alternate interior angles)
So, ∠ABO = ∠CDO   ...(2) 

In △ AOB and △ COD,
∠ DCO ≅ ∠BAO     .....(Pair of alternate interior angles)(1)
∠ CDO ≅ ∠ABO     .....(Pair of alternate interior angles)(2)

side DC = side AB ....(parallel side)

by ASA congruency condition,

△ AOB ≅ △ COD.

This gives, AO = CO and BO = DO.....(C.P.C.T)

Hence proved.

Example

In given Fig, HELP is a parallelogram. (Lengths are in cms). Given that OE = 4 and HL is 5 more than PE? Find OH.

If OE = 4 then OP also is 4. .......(The diagonals of a parallelogram bisect each other.)

So, PE = 8,

Therefore, HL = 8 + 5 = 13

Hence, OH = `1/2 xx 13` = 6.5 cms.

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