Q. 84.0( 18 Votes )
Given: AO = OB , DO = OC
To prove: AC=BD and AC||BD
It is given that, O is the midpoint of each of the line segments AB and CD.
This implies that AO = OB and DO = OC
Here line segments AB and CD are concurrent.
∠AOC = ∠BOD …. As they are vertically opposite angles.
Now in ∆AOC and ∆BOD,
AO = OB,
OC = OD
Also, ∠AOC = ∠BOD
Hence, ∆AOC ≅ ∆BOD … by SAS property of congruency
AC = BD … by cpct
∴ ∠ACO = ∠BDO … by cpct
But ∠ACO and ∠BDO are alternate angles.
∴ We conclude that AC is parallel to BD.
Hence we proved that AC=BD and AC||BD
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