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.
image

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,
image
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.

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