If the diagonals of a quadrilateral bisect each other then quadrilateral is a parallelogram. Prove it.
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Step-by-step explanation:
Answer
ABCD is an quadrilateral with AC and BD are diagonals intersecting at O.
It is given that diagonals bisect each other.
∴ OA=OC and OB=OD
In △AOD and △COB
⇒ OA=OC [ Given ]
⇒ ∠AOD=∠COB [ Vertically opposite angles ]
⇒ OD=OB [ Given ]
⇒ △AOD≅△COB [ By SAS Congruence rule ]
∴ ∠OAD=∠OCB [ CPCT ] ----- ( 1 )
Similarly, we can prove
⇒ △AOB≅△COD
⇒ ∠ABO=∠CDO [ CPCT ] ---- ( 2 )
For lines AB and CD with transversal BD,
⇒ ∠ABO and ∠CDO are alternate angles and are equal.
∴ Lines are parallel i.e. AB∥CD
For lines AD and BC, with transversal AC,
⇒ ∠OAD and △OCB are alternate angles and are equal.
∴ Lines are parallel i.e. AD∥BC
Thus, in ABCD, both pairs of opposite sides are parallel.
∴ ABCD is a parallelogram.
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Answer:
opposite
Step-by-step explanation:
since opposite sides are parallel