Question

8In the adjoining figure, ABCD is a
parallelogram and X, Y are the points
on the diagonal BD such that DX= BY
Prope that
CXAY is a parallelogram.
(w) LADXOCBY and AABY ACDX, and
fi) AX=CY and CX = AY​

Answer 1

ANSWER:-

Given:

In the adjoining figure, ABCD is a parallelogram and X, Y are the points on the diagonal BD such that DX= BY.

To prove:

⏺️AXCY is a parallelogram.

⏺️AX = CY

⏺️CX= AY

⏺️∆AYB = ∆CXD

⏺️∆AXD= ∆CYB

Proof:

In ∆AXD and ∆CYB,

∠ADX=∠CBY

[alternate interior angles for BC||AD]

AD= CB [opposite sides of ||gm ABCD]

DX= BY [given]

=)∆AXD≅∆CYB[using SAS congruence]

=) AX= CY [by c.p.c.t]

Now,

In ∆AYB and ∆CXD,

∠ABY=∠CDX

[alternate interior angles for AB||CD]

AB= CD [opposite sides of ||gm ABCD]

DX = BY [given]

=)∆AYB≅∆CXD [Using SAS congruence]

=) AY = CX [by c.p.c.t]

Since opposite sides are quadrilateral AXCY are equal to each other.

Therefore,

AXCY is a parallelogram. [Proved]

Hope it helps ☺️