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Question

ABCD is a parallelogram. The sides AB and AD are produced to E and F respectively, such that AB=BE and AD=DF.

Prove that BECDCF.


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Solution

Step 1: Drawing the diagram:

ABCD is a parallelogram which means ADBC and ABCD.

Sides AB and AD are produced to E and F, such that AB=BE and AD=DF.

Step 2: Proving BECDCF:

Corresponding angles for ABCD,

BAD=CDF

Corresponding angles for ADBC,

BAD=CBE

Therefore, CDF=CBE……………………….(1)

ABCD is a parallelogram, so we know that,

AD=BC and AB=CD……….(opposite sides of parallelogram are equal)

Given,

AB=BE and AD=DF

DF=BC…………………….. (2)

Similarly,

CD=BE…………………………(3)

In, DCF and BEC

CD=EBFromequation(3)CDF=EBCFromequation(1)DF=BCFromequation(2)

Therefore, by SAS congruency criteria,

BECDCF

Hence proved.


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