Question

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

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Solution

To prove: AC=BD and AC∥BD In △AOC and △BOD, we have: OA=OB (O is the midpoint)∠AOC=∠BOD (Vertically opposite angles) OC=OD (O is the midpoint)∴△AOC≅△BOD (By SAS congruency criterion)Also, ∠CAO=∠OBD (CPCT)and, AC=BD (CPCT)AC and BD is cut by a transversal AB, such that the alternate angles are equal i.e. ∠CAO=∠OBDSo, AC∥BD and AC=BD (proved above)Hence, proved.

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