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In triangles AOP and QOB, we have
AO = BO (O, the midpoint of AB);
∠AOP=∠BOQ, (vertically opposite angles);
PO=OQ, (O, the midpoint of PQ)
So by SAS postulate we have
△AOP≅△BOQ
Hence, AP = BQ, as they are corresponding parts of congruent triangles
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