Question

In figure ABCD is a parallelogram. AX and CY bisects angles A and C. prove
that AYCX is a parallelogram.

Answer 1

Answer:

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.

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Step-by-step explanation:

Answer 2

Answer:



Since opposite angles are equal in a parallelogram . Therefore , in parallelogram ABCD , we have 

∠A = ∠C 

⇒  1 / 2 ∠A = 1 / 2  ∠C 

⇒ ∠1 = ∠2 ---- i)

[∵ AX and CY are bisectors of ∠A  and ∠C respectively]

Now, AB | |  DC and the transversal CY intersects them. 

∴ ∠2 and ∠3 ---- ii) [∵ alternate interior angles are equal ]

From (i) and (ii) , we have 

∠1 and ∠3

Thus , transversal AB intersects AX and YC at A and Y such that  ∠1 = ∠3 i.e. corresponding angles are equal .

∴ AX | | CY .