Option B:
[tex]\overline{A B}=\overline{D E}[/tex]
Solution:
In the given figure [tex]\triangle A B C \cong \triangle D E C[/tex].
If two triangles are similar, then their corresponding sides and angles are equal.
By CPCTC, in [tex]\triangle A B C \ \text{and}\ \triangle D E C[/tex],
[tex]\overline{AB }=\overline{DE}[/tex] – – – – (1)
[tex]\overline{B C}=\overline{EC}[/tex] – – – – (2)
[tex]\overline{ CA}=\overline{CD}[/tex] – – – – (3)
[tex]\angle ACB=\angle DCE[/tex] – – – – (4)
[tex]\angle ABC=\angle DEC[/tex] – – – – (5)
[tex]\angle BAC=\angle EDC[/tex] – – – – (6)
Option A: [tex]\overline{B C}=\overline{D C}[/tex]
By CPCTC proved in equation (2) [tex]\overline{B C}=\overline{EC}[/tex].
Therefore [tex]\overline{B C}\neq \overline{D C}[/tex]. Option A is false.
Option B: [tex]\overline{A B}=\overline{D E}[/tex]
By CPCTC proved in equation (1) [tex]\overline{AB }=\overline{DE}[/tex].
Therefore Option B is true.
Option C: [tex]\angle A C B=\angle D E C[/tex]
By CPCTC proved in equation (4) [tex]\angle ACB=\angle DCE[/tex].
Therefore [tex]\angle A C B\neq \angle D E C[/tex]. Option C is false.
Option D: [tex]\angle A B C=\angle E D C[/tex]
By CPCTC proved in equation (5) [tex]\angle ABC=\angle DEC[/tex].
Therefore [tex]\angle A B C\neq \angle E D C[/tex]. Option D is false.
Hence Option B is the correct answer.
[tex]\Rightarrow\overline{A B}=\overline{D E}[/tex]
