Question

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

Answer 1

Given:-

• AX is bisector of $\rm{\angle{A}}$

• CY is the Bisector of $\rm{\angle{C}}$

To Prove :-

• AXCY is a Parallelogram.

Concept Used:-

• Alternate Angle Property

Now,

→ ABCD is a Parallelogram So,

→ $\rm{\angle{A} = \angle{C}}$

→ Halves of the angle will be also equal because AX and CY is a Bisector respectively.

Therefore,

→ $\rm{ \angle{ 1} = \angle{ 2}}$

Also,

→ AB || CD and AX is a Transversal.

So,

→ $\rm{\angle{ 2 } = \angle{ 3 }}$

But, $\rm{ \angle{ 1} = \angle{ 2}}$

Therefore,

→ $\rm{ \angle{ 1} = \angle{ 3}}$

Through this we get AX || CY

Now,

In ∆DAX & ∆CBY

→ AD = BC ( ABCD is a Parallelogram)

→ $\rm{\angle{B} = \angle{C}}$ ( opposite angles are equal )

→ $\rm{\angle{DAX} = \angle{YCB}}$ ( Alternate interior angle )

Hence, ∆DAX ≈ ∆CBY by A - A - S Congurency Criteria

Therefore,

→ AX = CY. ( C.P.C. T )

So, AXCY is a Parallelogram because Same side is equal and Parallel.

Answer 2

Answer:

Step-by-step explanation:

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)

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Thank You!