Question

Given: QM is the angle bisector of ∠LMP QM is the angle bisector of ∠PQL Prove: LM ≅PQ

Answer 1

Explanation:

∠LMQ = ∠QMP QM is the angle bisector of ∠LMP

∠PQM = ∠MQL QM is the angle bisector of ∠PQL

MQ = QM common

MQP ≅ MQL ASA

LP ⊥ MQ angle bisector theorem

LMPQ is rhombus property of rhombus

LM ≅ PQ property of rhombus

Answer 2

Explanation:

QM is the angle bisector of ∠LMP

∠LMQ = ∠QMP

QM is the angle bisector of ∠PQL

∠PQM = ∠MQL

MQ = QM as common

By ASA, triangle MQP ≅ MQL

LM = PM and LQ = PQ as they are same side of congruent triangles

Triangle LPQ and LPM are isosceles

By angle bisector theorem, LP is perpendicular to MQ

By properties of rhombus, the two diagonals are perpendicular proves that LMPQ is a rhombus.

LM ≅ PQ