In figure, two lines $A B$ and $C D$ intersect each other at the point $O$ such that $B C ∥ D A$ . Show that $O$ is the mid-point of both the line-segments $A B$ and $C D$ .

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Solution

$\begin{matrix} I n Δ C O B & Δ A O D (v e r t i c a l l y o p p o s i t e a n g e) ∠ C O B = ∠ A O D (a l t e r n a t e i n t e r i o r a n g l e) A B : B C (g i v e n) ∴ Δ C O B ≅ Δ A O D (A S A c o n g u r e n c y) ∴ O B : O A (C P C T) a n d O C = O D (C P C T) a n d h e n c e O i s t h e m i d p o i n t . \end{matrix}$

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