Question

to show that \\( \triangle fgh \cong \triangle jkl \\) by sas, what additional information is needed? check all that apply. \\( \square \overline{fh} \cong \overline{jl} \\) and \\( \overline{fg} \cong \overline{jk} \\) \\( \square \overline{fh} \cong \overline{jl} \\) and \\( \overline{hg} \cong \overline{lk} \\) \\( \square \angle g \cong \angle k \\) and \\( \overline{fh} \cong \overline{jl} \\) \\( \square \angle g \cong \angle k \\) and \\( \overline{gh} \cong \overline{kl} \\) \\( \square \angle g \cong \angle k \\) and \\( \overline{fg} \cong \overline{jk} \\)

Explanation:

Step1: Recall the SAS (Side - Angle - Side) congruence criterion

For two triangles $\triangle A_1B_1C_1$ and $\triangle A_2B_2C_2$, $\triangle A_1B_1C_1\cong\triangle A_2B_2C_2$ by SAS if $A_1B_1 = A_2B_2$, $\angle B_1=\angle B_2$, and $B_1C_1 = B_2C_2$ (i.e., two sides and the included angle of one triangle are congruent to the corresponding two sides and included angle of the other triangle).

Step2: Analyze each option

• Option 1: $\overline{FH}\cong\overline{JL}$ and $\overline{FG}\cong\overline{JK}$

The angle between $\overline{FH}$ and $\overline{FG}$ in $\triangle FGH$ is $\angle F$, and the angle between $\overline{JL}$ and $\overline{JK}$ in $\triangle JKL$ is $\angle J$. We don't know if $\angle F\cong\angle J$, so this does not satisfy SAS.

• Option 2: $\overline{FH}\cong\overline{JL}$ and $\overline{HG}\cong\overline{LK}$

The angle between $\overline{FH}$ and $\overline{HG}$ in $\triangle FGH$ is $\angle H$, and the angle between $\overline{JL}$ and $\overline{LK}$ in $\triangle JKL$ is $\angle L$. We don't know if $\angle H\cong\angle L$, so this does not satisfy SAS.

• Option 3: $\angle G\cong\angle K$ and $\overline{FH}\cong\overline{JL}$

$\overline{FH}$ is not adjacent to $\angle G$ in $\triangle FGH$, and $\overline{JL}$ is not adjacent to $\angle K$ in $\triangle JKL$. So, this does not satisfy the "side - angle - side" structure.

• Option 4: $\angle G\cong\angle K$ and $\overline{GH}\cong\overline{KL}$

In $\triangle FGH$ and $\triangle JKL$, if $\angle G\cong\angle K$, $\overline{GH}\cong\overline{KL}$, and we assume the other pair of sides (adjacent to the given angles) $\overline{FG}\cong\overline{JK}$ (by the structure of the triangles and the need for SAS). The sides $\overline{GH}$ and $\overline{FG}$ with included angle $\angle G$ in $\triangle FGH$ and $\overline{KL}$ and $\overline{JK}$ with included angle $\angle K$ in $\triangle JKL$ satisfy SAS ($\overline{GH}\cong\overline{KL}$, $\angle G\cong\angle K$, $\overline{FG}\cong\overline{JK}$ as a result of triangle correspondence).

• Option 5: $\angle G\cong\angle K$ and $\overline{FG}\cong\overline{JK}$

In $\triangle FGH$ and $\triangle JKL$, $\overline{FG}$ and $\overline{GH}$ with included angle $\angle G$ in $\triangle FGH$ and $\overline{JK}$ and $\overline{KL}$ with included angle $\angle K$ in $\triangle JKL$. Since $\angle G\cong\angle K$, $\overline{FG}\cong\overline{JK}$, and $\overline{GH}\cong\overline{KL}$ (by the structure of the triangles and the need for SAS), this satisfies the SAS congruence criterion.

Answer:

$\angle G\cong\angle K$ and $\overline{GH}\cong\overline{KL}$; $\angle G\cong\angle K$ and $\overline{FG}\cong\overline{JK}$