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Theorem - 2 : The locus of a point equidistant from two intersecting straight lines consists of a pair of straight lines which bisect the angle between the given lines.

Theorem is proved in two parts

Part – I

Every point which his eqinidistant from two intersecting lines, lies on the bisector of the angle between the given lines.

Given : - Two lines AB and CD intersect at O. P is a point such that PM = PN where

To Prove : - P lies on the bisector of

Construction : - O and P is joined

Proof : -

Part – II

Conversely, every point on the angle bisector of two intersecting lines, is equidistant from the lines.

Given : - Two lines AB and CD intersect at O. l bisects and P is any point on l.

To Prove : - P is equidistant from AB and CD.

Construction : - PM perpendicular to AB and PN perpendicular to CD are drawn.

Proof : -

Hence every point equidistant from two intersecting lines AB and CD lies on the bisector of and every point on the bisector of is equidistant from AB and CD.