Question

in the given figure abcd is a parallelogram.ax and cy bisect angles and c. prove that aycx is a parallelogram

Answer 1

Given: ab c d is a parallelogram in which ax bisects angle a and cy bisects angle c.

To prove: a y c x is a parallelogram.

proof: ax bisects angle a.

So,∠1=∠2

c y bisects ∠c.

∠3=∠4

As, ab c d is a parallelogram.

Opposite angles are equal.

∠a = ∠ c

⇒∠1 +∠2=∠3+∠4

⇒2∠2=2∠3

⇒∠2=∠3

also, a y║c x   →[As a b║ c d,∴a y║c x  As a y and c x are part of ab and c d.]

∠3+∠5=180°and ∠2+∠6=180°[When lines are parallel, sum of supplementary angles is 180°]

∴ ∠3+∠5=∠2+∠6

But→ ∠2=∠3

∴∠5=∠6

As in quadrilateral a x c y , ∠2=∠3 and∠5=∠6.But these are pair of opposite angles of quadrilateral a x c y . So quadrilateral a x c y is a parallelogram.