Question

AX and CY are respectively the bisectors of opposite angles of A and C of a parallelogram ABCD . Show that AX is parallel to CY?

Answer 1

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 parallelogram are equal)

We know that halves of equals are equal, therefore:

⇒∠A/2 = ∠C/2

⇒ ∠1 = ∠2 → Eq(1) (AX and CY bisects ∠A and ∠C)

AB ║CD and CY is the transversal.

(Opposite sides of a ║gm are parallel to each other)

∴ ∠2 = ∠3 → Eq(2) (Alternate Interior angles)

From Eq(1) and Eq(2):

∠1 = ∠3 [Things equal to the same thing are equal to one another]

∴ AX ║ CY

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

Hence proved.

Answer 2

Answer:

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