Math, asked by aditya45676, 11 months ago

ABCD is a parallelogram and line-segments AX, CY bisect the angles A and C respectively. Prove that, AX || CY.

Answers

Answered by Anonymous
25

Given: A ||gm ABCD and line segments AX, CY bisect the angles A and C bisect ∠A and ∠C respectively.

To prove: AX || CY

Proof:

∠A = ∠C (opposite ∠s of a ||gm)

∴ 1/2 ∠A = 1/2∠C

i.e. ∠XAY = ∠XCY (AX and CY are bisectors of ∠s A and C respectively)

Now,

AB || DC and CY intersect them

∴ ∠XAY + ∠AXC = 180° (Co-interrior angles) ...(i)

AB || DC and CY intersects them

∴ ∠XCY + ∠AYC  = 180° (Co-interrior angles).....(ii)

From equations (i) and (ii) we get that,

∠XAY + ∠AXC = ∠XCY + ∠AYC

⇒ ∠XAY + ∠AXC = ∠XAY + ∠AYC (∵ ∠XYC = ∠XAY, PROVED)

∴ ∠AXC = ∠AYC

Such that, opposite angles of quadrilateral AYCX are equal.

Hence, AYCX is ||gm ⇒ AX || CY

Answered by Anonymous
16

Answer:

check the attachment (◔‿◔)

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