in the given figure, If $A B ∥ D C$ and $P$ is the midpoint of $B D$ , prove that $P$ is also the midpoint of $A C$ .

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Solution

Given:- $A B | | D C$
To prove : - $P$ is midpoint of $A C$
Proof :- $P$ is midpoint of $D B$ .
Here $D P = B P$ .
$∴ ∠ D C P = ∠ B A P$ .
and $∠ D P C = ∠ A P B$
$D P = B P$
$∴$ now $Δ D P C ≅ Δ A P B$
$∴ ∠ C D P = ∠ P B A$
$∴ A P = P C$
$∴ A P + P C = A C$ [we know]
$2 A P = A C$
$[A P = \frac{A C}{2}]$
$∴ P$ is midpoint of $A C$

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