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If a point lies on the bisector of an angle, then the point is equidistant from the two sides of the angle.
Based on the figure, the following conditional statement holds true.
{ c ADbisects∠ BAC DB⊥AB DC⊥AC . ⇒ DB=DC
Let D be a point lying on the bisector of the angle. Also, let DB and DC be the distances from D to the sides of the angle. Recall that the distance from a point to a line is perpendicular to the line.
Since AD bisects ∠ BAC, by the definition of an angle bisector it can be said that ∠ BAD and ∠ CAD are congruent angles. Furthermore, ∠ ABD and ∠ ACD are both right angles. Therefore, they are also congruent angles. ∠ BAD≅∠ CAD and ∠ ABD≅∠ ACD By the Reflexive Property of Congruence, AD is congruent to itself.
Because two angles and a non-included side of △ ADB are congruent to two angles and the corresponding non-included side of △ ADC, the triangles are congruent by the Angle-Angle-Side Congruence Theorem. △ ADB≅△ ADC Corresponding parts of congruent triangles are congruent. Therefore, DB is congruent to DC. Congruent segments have equal measures. DB=DC This means that D is equidistant from the rays of ∠ BAC.