Question

In the adjoining figure ABCD is a parallelogram and x y are the points on diagonal BD such that BD equal BY prove that 1) CXAY is a parallelogram 2) △ADX≅CBY and △ABY≅CDX 3)AX=CY and CX = AY​

Answer 1

$\bf\underline{\underline{\pink{Question:-}}}$

✭ In the adjoining figure ABCD is a parallelogram and x y are the points on diagonal BD such that BD equal BY.

Prove that:

1) CXAY is a parallelogram

2) △ADX≅CBY and △ABY≅CDX

3) AX=CY and CX = AY

$\bf\underline{\underline{\blue{Given:-}}}$

✭ ABCD is a parallelogram and x y are the points on diagonal BD such that BD equal BY

$\bf\underline{\underline{\green{ToProve:-}}}$

✭CXAY is a parallelogram

✭△ADX≅CBY and △ABY≅CDX

✭ AX=CY and CX = AY

$\bf\underline{\underline{\red{Proof:-}}}$

✭ CXAY is a parallelogram

Join AC, meeting BD at O.

Since the diagonals of a parallelogram bisect each other, we have OA = OC and OD = OB.

Now, OD = OB and OD = OB.

==> OD – DX = OB – BY => OX = OY.

Now, OA = OC and OX =OY

CXAY is a quadrilateral whose diagonals bisect each other.

∴ CXAY is a ||gm.

✭ △ADX≅CBY and △ABY≅CDX and AX=CY and CX = AY

In △ADX and △CBY, we have

AD = BC⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀(opp.sides of a ∥gm )⠀⠀⠀⠀⠀⠀⠀

∠ADX = ∠CBY⠀⠀⠀⠀⠀⠀⠀⠀⠀(alt.interior ∠s)⠀⠀⠀⠀⠀⠀⠀

DX = BY⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀(given)

∴△ADX≅CBY and △ABY≅CDX (SAS- criterion).

And so, AX=CY ⠀⠀⠀⠀⠀⠀⠀⠀⠀(C.P.C.T)

similarly,△ABY≅CDX and AX=CY and so CX = AY (C.P.C.T)