In the adjoining figure, P is a point in the interior of ∠AOB. If PL ⊥ OA and PM ⊥ OB such that PL=PM, show that OP is the bisector of ∠AOB
Given: P is a point in the interior of ∠AOB and PL ⊥ OA and PM ⊥ OB such that PL=PM
To prove: ∠POL = ∠POM
Proof:
In ∆OPL and ∆OPM, we have
∠OPM = ∠OPL = 90° …given
OP = OP …common side
PL = PM … given
Thus by Right angle hypotenuse side property of congruence,
∆OPL ≅ ∆OPM
Hence, we know that, corresponding parts of the congruent triangles are equal
∴ ∠POL = ∠POM
Ie. OP is the bisector of ∠AOB