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Ans:

Given: ABCD is a parallelogram where AX bisects A and CY bisects C.

To Prove: AX||CY

Proof: Since opposite angles are equal in a parallelogram, we get,
A = C

or A = C
[since, AX and AY bisects A and C respectively] (i)
AB||DC and CY is a transversal.
(ii) [alternate interior angles]
Therefore, [using (i) and (ii)]
But these are corresponding angles.
AX||CY [converse of corresponding angles theorem]
Example:

ABCD is a rhombus with one diagonal equal to 18 cm and length of side equal to 15 cm.
Find the area of rhombus.

Rhombus ABCD with AB = BC = CD = AD
In AOB
2
2
2
AB = OA + OB (Diagonals of rhombus are perpendicular to each other)
2
2
2
15 = 9 + OA
2
2
2
15 - 9 = OA
2
225 - 81 = OA
2
144 = OA
OA = 12
Therefore, AC = 24 cm
Area of rhombus = d 1 d 2
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