Blyloloos #72 PROVING A THEOREM Complete the proof of the Incenter Theorem. 520 Given AABC. AD bisects CAB, BD bisects ZCBA, DE LAB, DF LBC, and DG CA Prove The angle bisectors intersect at D which is equidistant from AB, BC, and CA B Because DE LAB DF IBC, and DG ICA, ZDFB. ZDEB, ZDEA, and ZDGA are Also, by the definition of F Because DE LAB DF 1 BC, and DG I CA. ZDFB, DEB, DEA, and ZDGA are Also, by the definition of ZDBFZDBE and ZDAEZDAG. In addition, DB DB and DADA by the So, ADFBADEB and ADEA ADGA by the Next, because corresponding parts of congruent triangles are congruent, DF DEDG.So point Dis DF DE and DG DE By the equidistant from AB, BC, and CA. Because D is equidistant from CA and CB by the point D is on the angle bisector of CACB So, the angle bisectors intersect atunnit point D - acute angles #congruent right angles 11 congruent triangles + angle bisector :: perpendicular bisector Reflexive Property of Congruence Symmetric Property of Congruence AAS Congruence Theorem # ASA Congruence Theorem Transitive Property of Congruence :: Converse of the Angle Bisector Theorem :: Angle Bisector Theorem