Question

In fig. 3.21, seg PQ and seg RS intersect at point O such that
seg OP $\cong$ seg OS
seg OR $\cong$ seg OQ
Show that seg PR $\cong$ seg SQ justify whether seg PR and seg SQ are parallel.

Answer 1

Given : seg OP $\cong$ seg OS ...... (1)
seg OR $\cong$ seg OQ ......(2)
In$△$POR and $△$SOQ,
seg OP $\cong$ seg OS (given)
$\angle$POR $\cong$ $\angle$SOQ (vertically-opposite angles)
seg OR $\cong$ seg OQ (given)
Therefore,$△$POR $\cong$ $△$SOQ (SAS congruency test)
Therefore,seg PR $\cong$ seg SQ (c.s.c.t)
Now in $△$POR and $△$SOQ,
$\angle$OPR = $\angle$OSQ and $\angle$ORP = $\angle$OQS (by c.a.c.t)
But $\angle$OPR $\ne$$\angle$OQS
Similarly $\angle$ORP $\ne$ $\angle$OSQ

Hence, PR is not parallel to SQ.