BD is a perpendicular bisector of AC. Which statement can NOT always be proven?

$∠ A D B ≅ ∠ C D B$

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$A D ≅ C D$

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$Δ A B D ≅ Δ C B D$

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$B D ≅ D C$

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Solution

The correct option is D
$B D ≅ D C$

$A D ≅ C D$ and $∠ A D B ≅ ∠ C D B$ as $B D$ is a perpendicular bisector.
So by using $S A S$ criterion we can prove that $Δ A B D ≅ Δ C B D$ .

$B D$ and $C D$ are not the corresponding sides and we do not have any other information about the sides.

Hence, it is not possible to prove that $B D ≅ D C$ .

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