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:
- Let △ABC and △DEF be the two triangles such that BC=EF,∠ACB=∠DFE, and ∠ABC=∠DEF.
- We need to prove that △ABC≅△DEF.
- 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]∴ Now, let us assume AC \neq DF .
Let us construct D' on the line FD such that D'F = AC and then join the point D' to the point E .
- As corresponding parts of congruent triangles are equal, we have
\angle ABC = \angle D'EF
But, \angle ABC = \angle DEF \space \text{[Given]}
\implies \angle D'EF = \angle DEF
This is possible only when D and D' coincide.
Hence, \bf {\triangle ABC \cong \triangle DEF}.