Question

Abcd is a quadrilateral in which ab=ad and bc=dc prove that ac is perpendicular bisector of bd

Answer 1

as ad=ab angleADB=ABD
And angleCBD=CDB
Adding - AngleADC=ABC
Opposite angle equalsABCD IS A PARALLELOGRM
AND ACIS PB OF BD

Answer 2

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Now,

△ ABC and. △ ADC

AB = AD ___(given)

CB = CD ____(given)

AC = AC _____(common side)

Now,

hence,

△ ABC ≈ △ ADC _____(by SSS rule)

so,

angle BAO = angle DAO ___(by CPCT)

here, ABD is a isosceles triangle as

AB = AD

SO,

angle ABO = angle ADO
___________(by base angle theorem)

in △ABO and △ADO

angle BAO = angle DAO___(as we proved)

AB = AD ____(given)

angle ABO = angle ADO
___________(by base angle theorem)

Hence,

△ABO ≈△ADO

SO,

OB = OD ________eq(1)

and

angle AOB = angle AOD _____eq(2)
________________(by CPCT)

AND,

angle AOB + angle AOD = 180 DEGREE

2×angleAOB = 180 DEGREE
_______________(by eq(2))

angle AOB = 90 DEGREE

SO,

angle AOB = angle AOD = 90 ___eq(3)

so from eq(1) and eq(3)

we get,

AC is the perpendicular bisector of BD.

___________________"PROVED"

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