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Tell whether the following statement is true or false:

In the adjoining figure, ABCD is a parallelogram and E is the midpoint of side BC. If DE and AB when produced meet at F, then AF = 2AB.

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True

False

answer is A.

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Detailed Solution

Given ABCD is a parallelogram and E is the midpoint of side BC.
For the triangles DEC and FEB,

∠ DCB = ∠ EBF

(Using alternate interior angle property)
CE = BE (given E is midpoint)

∠ CED = ∠ BEF

(Using vertically opposite angle property)
Therefore, triangles DEC and FEB are congruent triangles using RHS postulate.
Now, as triangles are congruent, so CD = BF.
But, as ABCD is a parallelogram, therefore CD = AB.
So, BF = AB. Now,

AF = AB + BF AF = AB + AB AF = 2AB

Hence, the given statement is true.
Therefore, option 1 is correct.

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