Question

B
F
Fig. 5 C
25. In quadrilateral ABCD, the diagonals intersect each other at E such that EA : EC = 3:2.
Is ABCD a parallelogram? Give reasons for your answer.
(2 marks)​

Answer 1

Answer:

Here, ABCD is a quadrilateral with diagonals AC and BDintersect at point  E.

⇒  ECAE=EDBE               [ Given ]

⇒  BEAE=EDEC         ---- ( 1 )

In △ABE and △CDE,

⇒  BEAE=EDEC              ] From ( 1 ) ]

⇒  ∠ABE=∠DEC              [ Vertically opposites angles ]

∴  △ABE∼△CDE           [ By SASsimilarity theorem ]

⇒  ∠EDC=∠EBA           [ Corresponding angles are equal. ]

∴  ∠BDC=∠ABD

Bu, this is a pair of alternate angles.

∴  AB∥DC

Thus, the quadrilateral ABCDis trapezium.