prove that if in two triangles two angles and the included sideof one triangle are equal to two angles and the included side of the others triangle ,then two triangles are congurent
Answers
Answer:
Step-by-step explanation:
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.
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