Question

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

Answer 1

Answer:

Step-by-step 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

Answer 2

Answer:

Step-by-step 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