In the given figure, $O$ is the midpoint of each of the line segments $A B$ and $C D .$ Prove that $A C = B D$ and $A C ∥ B D .$

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Solution

To prove: $A C = B D$ and $A C ∥ B D$
In $△ A O C$ and $△ B O D,$ we have:
$O A = O B$ ( $O$ is the midpoint)
$∠ A O C = ∠ B O D$ (Vertically opposite angles)
$O C = O D$ ( $O$ is the midpoint)
$∴ △ A O C ≅ △ B O D$ (By $S A S$ congruency criterion)
Also, $∠ C A O = ∠ O B D$ ( $C P C T$ )
and, $A C = B D$ ( $C P C T$ )
$A C$ and $B D$ is cut by a transversal $A B,$ such that the alternate angles are equal i.e. $∠ C A O = ∠ O B D$
So, $A C ∥ B D$
and $A C = B D$ (proved above)
Hence, proved.

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Question 13
In the given figure, l||m and line segments AB, CD and EF are concurrent at point P. Prove that
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In the given figure, O is the midpoint of each of the line segments AB and CD. Prove that AC=BD and AC∥BD.

In the given figure, $O$ is the midpoint of each of the line segments $A B$ and $C D .$ Prove that $A C = B D$ and $A C ∥ B D .$