CBSE Class 9 Answered

Given: Two triangles ABC and DEF such that B = E, C = F

and BC = EF. To prove: ABC DEF

Proof: Case I: If AB = DE then in ABC and DEF, AB = DE [by supposition] BC = EF [given] and B = E [given] Thus, ABC DEF [SAS criterion] Case II: If AB < DE Take a point G on ED such that EG = AB. Join GF. In ABC and GEF, we have AB = GE [by supposition] BC = EF [given] B = E [given] Thus, ABC GEF [SAS criterion] ACB = GFE [corresponding parts of congruent triangles are equal] But ACB = DFE [given] GFE = DFE, This is only possible when FG coincides with FD or G coincides with D. AB must be equal to DE and hence, ABC DEF (by SAS) Case III: If AB > ED With a similar argument (as in case II), we may conclude that ABC DEF (by SAS) Thus, ABC DEF.