Question

If MP=NQ prove that triangle S OMP and ONQ are congruent and hence prove that MN and PQ bisect each other at O.​

Answer 1

Given data

1.   MP=NQ

     ∠MPO =∠NQO =90°

2.   In ΔMPO and ΔNQO

     ∠MPO =∠NQO       (because each angle is 90°)

     ∠MOP =∠NOQ       (vertical opposite angle)

      MP= QN                 (Given)

      ΔMPO ≅ ΔNQO    (by AAS method)

3.   So

      MO = ON                (by cpct)     ...1)

      and

      PO = OQ                 (by cpct)    ...2)

      From equation 1) and equation 2), it is clear that line PQ  

      and line MN bisect each other.