If two angles and the included side of one triangle are equal to the corresponding two angles and the included side of the other triangle, then prove that the two triangles are congruent.


Answer:


Step by Step Explanation:
  1. Let ABC and DEF be the two triangles such that BC=EF,ACB=DFE, and ABC=DEF.
      A C B D F E
  2. We need to prove that ABCDEF.
  3. Let us assume that AC=DF.

    In ABC and DEF, we have AC=DF  [Just assumed]BC=EF  [By step 1]ACB=DFE  [By step 1]ABCDEF  [By SAS-Criterion] Now, let us assume ACDF.

    Let us construct D' on the line FD such that DF=AC and then join the point D to the point E.
      A C B D F ED' D'
    In ABC and DEF, we have AC=DF  [By construction]BC=EF  [By step 1]ACB=DFE  [By step 1]ABCDEF  [By SAS-criterion]
  4. As corresponding parts of congruent triangles are equal, we have ABC=DEF But, ABC=DEF [Given]

    DEF=DEF

    This is possible only when D and D coincide.

    Hence, ABCDEF.

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