Given:
[tex]\overline{BD} \cong \overline{BE}[/tex]
DE bisects [tex]\overline{AB}[/tex].
DE bisects [tex]\overline{BC}[/tex].
To prove:
[tex]\overline{AB} \cong \overline{BC}[/tex]
Proof:
DE bisects [tex]\overline{AB}[/tex]
Then, [tex]AD=BD[/tex]. ...(i)
DE bisects [tex]\overline{BC}[/tex].
Then, [tex]BE=CE[/tex]. ...(ii)
We have,
[tex]\overline{BD} \cong \overline{BE}[/tex] ...(iii)
Using (i), (ii) and (iii), we get
[tex]\overline{AD} \cong \overline{CE}[/tex] ...(iv)
Adding (iii) and (iv), we get
[tex]\overline{BD}+\overline{AD} \cong \overline{BE}+\overline{CE}[/tex]
[tex]\overline{AB} \cong \overline{BC}[/tex]
Hence proved.
