Question # Decide whether enough information is given to prove that the triangles are congruent using the SAS Congruence Theorem.12210203322.jpg

Congruence
ANSWERED Decide whether enough information is given to prove that the triangles are congruent using the SAS Congruence Theorem.  2020-12-17

Step 1
To make the triangle congruent by using the SAS congruence,
First naming the edges and vertex of the triangle, Step 2
By using sum of interior angle property in $$\displaystyle\triangle{A}{B}{C}{\quad\text{and}\quad}\triangle{D}{E}{F}$$,
$$\displaystyle\angle{A}{B}{C}={180}^{{\circ}}-{\left(\angle{A}{B}{C}+\angle{B}{A}{C}\right)}$$
$$\displaystyle\angle{D}{E}{F}={180}^{{\circ}}-{\left(\angle{E}{D}{F}+\angle{D}{F}{E}\right)}$$
it is given that,
$$\displaystyle\angle{A}{C}{B}=\angle{E}{D}{F}={90}^{{\circ}}$$ (given)...(i)
$$\displaystyle\angle{B}{A}{C}=\angle{D}{F}{E}{\left({g}{i}{v}{e}{n}\right)}$$...(ii)
Adding equation (i) and (ii) and subtracting them form $$\displaystyle{180}^{{\circ}}$$ in both sides,
$$\displaystyle{180}^{{\circ}}-{\left(\angle{A}{C}{B}+\angle{B}{A}{C}\right)}={180}^{{\circ}}-{\left(\angle{E}{D}{F}+\angle{D}{F}{E}\right)}$$
$$\displaystyle\angle{A}{B}{C}=\angle{D}{E}{F}$$
Now, in $$\displaystyle\triangle{A}{B}{C}{\quad\text{and}\quad}\triangle{D}{E}{F}$$,
AB=EF(given)
$$\displaystyle\angle{A}{B}{C}=\angle{D}{E}{F}$$(Proved above)
BC=DE(given)
$$\displaystyle\triangle{A}{B}{C}\stackrel{\sim}{=}\triangle{D}{E}{F}$$ (By SAS Congruence Theorem)
Yes, the provided information is enough to prove the triangle congruent by SAS.