Ask question
Question

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

Congruence
ANSWERED
asked 2020-12-16

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

## Answers (1)

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.

expert advice

...