Question

in the adjoining figure, abcd is a parallelogram and ax is parallel to cy prove that ax =cy and axcy is a parallelogram

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Answer 1

Question:

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

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)

∠A/2 = ∠C/2  (Halves of equals are equal)

∴ ∠1 = ∠2 (AX and CY bisects A and C)

We know that:

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

∴ ∠2 = ∠3  

But ∠1 = ∠2  as well.

Therefore from the above statements, we can conclude that:

⇒ ∠1 = ∠3  [Thins 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.