Q8 of 28 Page 95

In the figure, ABCD is a parallelogram. AB is produced to E such that AB = BE. AD produced to F such that AD = DF. Show that ∆FDC ≡ ∆CBE.

Given: Parallelogram ABCD and AB = BE and AD = FD


To prove: Δ FDC ≡ ΔCBE


Construction: Join DB



Proof:


We know that,


AB = DC [ opposite sides of parallelogram]


BE = DC [AB = BE, because B is the midpoint of AE]


Similarly,


AD = BC [ opposite sides of parallelogram]


DF = BC [ AD = DF, because B is the midpoint of AE]


Now, AD||BC and AB


∠ A = ∠ B [corresponding angles] …(1)


Now, AB||CD and AD


∠ A = ∠ D [corresponding angles] …(2)


∴ ∠ B = ∠ D (From 1 and 2)


In Δ FDC and Δ CBE


FD = CB [Proved Above]


DC = BE [Proved Above]


∠ D = ∠ B [Proved Above]


Thus, Δ FDC ≡ Δ CBE


Hence Proved.


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