and ( ) Adding to both sides Cancelling AC on both sides Our assumption that DE is not parallel to BC is wrong. Hence proved Theorem : Angle Bisector Theorem Statement The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the corresponding sides containing the angle. Proof Given : In D ABC, AD is the internal bisector To prove : AB AC BD CD Construction : Draw a line through C parallel to AB . Extend AD to meet line through C at E D F E Fig.
. E D Fig. . Geometry No Statement Reason .
∠ = ∠ = ∠ AEC BAE ∠ ABD = ∠ ECD = ∠ Two parallel lines cut by a transversal make alternate angles equal. . D ACE is isosceles AC = CE … ( ) In ∆ ACE CAE CEA , ∠ = ∠ . D D ABD ECD AB CE BD CD By AA Similarity .
AB AC BD CD From ( ) AC CE Hence proved. Activity Step : Take a chart and cut it like a triangle as shown in Fig. . (a).
Step : Then fold it along the symmetric line AD . Then C and B will be one upon the other. Step : Similarly fold it along CE, then B and A will be one upon the other. Step : Similarly fold it along BF, then A and C will be one upon the other.
Find AB, AC, BD, DC using a scale. Find AB AC BD DC check if they are equal? In the three cases, the internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the corresponding sides containing the angle. What do you conclude from this activity?
Theorem : Converse of Angle Bisector Theorem Statement If a straight line through one vertex of a triangle divides the opposite side internally in the ratio of the other two sides, then the line bisects