Question

In parallelogram ABCD, AX is bisector of angleA
and CY is the bisector of angleC. Prove that AXCY
is a parallelogram.
​.​

Answer 1

$\huge \cal \underline \color{red}★A᭄ɴsᴡΣʀ࿐$

Given,

• ABCD is a Parallelogram

• AX is the bisector of ∠A
• CY is the bisector of ∠C

To Prove

AX ║ CY

Proof

ABCD is a parallelogram

∴ ∠A = ∠C(Opp. angles of a paralleogram are equal)

∠A = ∠C (Halves of equals are equal)

∴ ∠1 = ∠2

(AX and CY bisects A and C)

AB ║CD and CY is the transversal (Parallel lines of a parallelogram)

∴ ∠2 = ∠3

But ∠1 = ∠2

⇒ ∠1 = ∠3

∴ AX ║ CY

(Corresponding angles 1 and 3 are equal, therfore the lines are parallel)

_______________________

Answer 2

Answer:

$\huge{\underline{\underline {\mathtt{\purple{A}\pink{N}\green{S}\blue{W}\red{E}\orange{R}}}}}$

Given,

ABCD is a Parallelogram

AX is the bisector of ∠A

CY is the bisector of ∠C

To Prove

AX ║ CY

Proof

ABCD is a parallelogram

∴ ∠A = ∠C(Opp. angles of a paralleogram are equal)

∠A = ∠C (Halves of equals are equal)

∴ ∠1 = ∠2

(AX and CY bisects A and C)

AB ║CD and CY is the transversal (Parallel lines of a parallelogram)

∴ ∠2 = ∠3

But ∠1 = ∠2

⇒ ∠1 = ∠3

∴ AX ║ CY

(Corresponding angles 1 and 3 are equal, therfore the lines are parallel)