In the figure, AC = DE = 7 cm. If BD = 4 cm, then CF equals

2 cm

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3 cm

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4 cm

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5 cm

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Solution

The correct option is C
4 cm

Construction: Extend ED to G, where it cuts AB. Similarly, extend AC to cut EF at H.
$∠ E D F = ∠ B C A$ [Given]
Given, $A B ∥ E F$ .
$∠ D E F = ∠ D G B \dots$ (1)
(Alternate interior angles)
Since $A C ∥ D E$ , we must have
$∠ D G B = ∠ B A C \dots$ (2) (Corresponding angles)
From (1) and (2),
$∠ D E F = ∠ B A C$
In $Δ A B C$ and $Δ E F D$ ,
$∠ A C B = ∠ E D F$ [Given]
$∠ B A C = ∠ D E F$ [Proven]
AC = DE [Given]
$⟹ Δ A B C ≅ Δ E F D$ (AAS congruency criteria)
So, BC = DF [CPCT]
$⟹$ BC - DC = DF - DC
$⟹$ BD = CF
$⟹$ CF = 4 cm [ $∵$ BD = 4 cm (given)]

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