Question

In the given figure, $O$ is the midpoint of each of the line segments $AB$ and $CD.$ Prove that $AC=BD$ and $AC\parallel BD.$

Answer 1

To prove: $AC=BD$ and $AC\parallel BD$

 In $△AOC$ and $△BOD,$ we have:

$OA=OB$ ($O$ is the midpoint)

$\angle AOC=\angle BOD$ (Vertically opposite angles)

$OC=OD$ ($O$ is the midpoint)

$\therefore △AOC\cong △BOD$ (By $SAS$ congruency criterion)

Also, $\angle CAO=\angle OBD$ ($CPCT$)

and, $AC=BD$ ($CPCT$)

$AC$ and $BD$ is cut by a transversal $AB,$ such that the alternate angles are equal i.e. $\angle CAO=\angle OBD$

So, $AC\parallel BD$

and $AC=BD$ (proved above)

Hence, proved.