In the figure PQ > PR, QM and RM are the bisectors of ∠Q and ∠R respectively. Prove that QM > RM.
Since QM and RM are angle bisectors of ∠Q and ∠R.
∴ ∠1 = ∠2 and ∠3 = ∠4
Now in ΔPQR, PQ > PR
... ∠PRQ > ∠PQR ...... (Angles opposite to larger side)
... ∠PRM + ∠MRQ > ∠PQM + ∠MQR
... ∠4 + ∠3 > ∠1 + ∠2
Þ 2 ∠3 > 2 ∠2
⇒ ∠3 > ∠2
... QM > RM.
∴ ∠1 = ∠2 and ∠3 = ∠4
Now in ΔPQR, PQ > PR
... ∠PRQ > ∠PQR ...... (Angles opposite to larger side)
... ∠PRM + ∠MRQ > ∠PQM + ∠MQR
... ∠4 + ∠3 > ∠1 + ∠2
Þ 2 ∠3 > 2 ∠2
⇒ ∠3 > ∠2
... QM > RM.
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