Question

PQRS is a parallelogram. PO and QO are respectively the angle bisectors to angle P and angle Q. line LOM is drawn parallel to PQ. Prove that i. PL=QM and ii. LO=MO

Answer 1

1-in 11gm pqrs
ps parallel qr
then,pl parallel qm
and,pq parallel lm(given)
hence pq is a 11gm
pl=qm


2-angle lpo=angle opq
but,angle lpo=angle opq(ALA)
angle lpo=angle lpo
pl=lo
similarly,pl=qm
so,lo=mo

Answer 2

Given:

PQRS is a parallelogram

PO and QO are respectively the angle bisectors to angle P and angle Q

LOM is parallel to PQ

To Prove:

i. PL=QM

ii. LO=MO

Solution:

Since PO bisects angle P,

⇒ ∠OPL = ∠OPQ       -(1)

Since PQ ║ LM and PO acts as a transversal,

⇒ ∠POL = ∠OPQ     (Alternate interior angles are equal)       - (2)

Equating (1) and (2), we get:

∠POL = ∠OPL

So, the opposite angles in ΔLOP are equal,

⇒ Their corresponding sides must be equal

⇒ PL = LO      - (3)

Similarly, since OQ is the angle bisector of angle Q,

We can prove that QM = MO       - (4)

Now, PQ ║ LM and PL ║ QM

⇒ PQML is also a prallelogram

⇒ PL = QM (Opposite sdes of a prallelogram are equal)

If PL = QM, using (3) and (4) we can say that

LO = MO

∴Hence, Proved