Question

# To prove: The similarity of \triangle BCD with respect to \triangle FED. Given information: Here, we have given that \overline{AC}\cong \overline{AE}\ and\ \angle CBD\cong \angle EFD

Similarity
To prove: The similarity of $$\displaystyle\triangle{B}{C}{D}$$ with respect to $$\displaystyle\triangle{F}{E}{D}$$.
Given information: Here, we have given that $$\displaystyle\overline{{{A}{C}}}\stackrel{\sim}{=}\overline{{{A}{E}}}\ {\quad\text{and}\quad}\ \angle{C}{B}{D}\stackrel{\sim}{=}\angle{E}{F}{D}$$

2021-04-18
Proof: AS, $$\displaystyle\overline{{{A}{C}}}\stackrel{\sim}{=}\overline{{{A}{E}}}$$,
$$\displaystyle\Rightarrow\angle{A}{C}{E}\stackrel{\sim}{=}\angle{A}{E}{C}$$ (Angles opposite to equal sides are equal)
Or, $$\displaystyle\angle{B}{C}{D}\stackrel{\sim}{=}\angle{F}{E}{D}$$
Now, In $$\displaystyle\triangle{B}{C}{D}\ {\quad\text{and}\quad}\ \triangle{F}{E}{D}$$
$$\displaystyle\angle{B}{C}{D}\stackrel{\sim}{=}\angle{F}{E}{D}$$ (Proved above)
$$\displaystyle\angle{C}{B}{D}\stackrel{\sim}{=}{E}{F}{D}$$ (Given)
$$\displaystyle\Rightarrow\triangle{B}{C}{D}\sim\triangle{F}{E}{D}$$ (By AA Similarity Rule)