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 SAS congruence rule

SAS requires two pairs of congruent sides, with the included angle (the angle between the two sides) congruent between the triangles.

Step2: Match marked angles/sides

In $\triangle FGH$, $\angle H$ is marked; in $\triangle JKL$, $\angle L$ is marked, so $\angle H \cong \angle L$. For SAS, we need the two sides forming these angles to be congruent.
For $\angle H$, the sides are $\overline{FH}$ and $\overline{HG}$. For $\angle L$, the sides are $\overline{JL}$ and $\overline{LK}$. So we need $\overline{FH} \cong \overline{JL}$ and $\overline{HG} \cong \overline{LK}$.
Alternatively, check the other marked angle pair: $\angle F \cong \angle J$. For $\angle F$, sides are $\overline{FG}$ and $\overline{FH}$; for $\angle J$, sides are $\overline{JK}$ and $\overline{JL}$. Wait, no—wait the other option: if we consider $\angle G \cong \angle K$, the sides forming $\angle G$ are $\overline{FG}$ and $\overline{GH}$; sides forming $\angle K$ are $\overline{JK}$ and $\overline{KL}$. So $\angle G \cong \angle K$ and $\overline{GH} \cong \overline{KL}$ and $\overline{FG} \cong \overline{JK}$? No, wait the option $\angle G \cong \angle K$ and $\overline{FG} \cong \overline{JK}$ and $\overline{GH} \cong \overline{KL}$? Wait no, the option $\angle G \cong \angle K$ and $\overline{FG} \cong \overline{JK}$: no, that would be if $\angle G$ is between $\overline{FG}$ and $\overline{GH}$, so we need $\overline{FG} \cong \overline{JK}$ and $\overline{GH} \cong \overline{KL}$ with $\angle G \cong \angle K$. Wait the option $\angle G \cong \angle K$ and $\overline{GH} \cong \overline{KL}$: no, that's one side and angle. Wait no, let's recheck:
Wait the first valid pair: $\overline{FH} \cong \overline{JL}$ and $\overline{HG} \cong \overline{LK}$ (matches the second option), because $\angle H \cong \angle L$ (marked), so SAS: two sides around the congruent angle.
Then the other valid option: $\angle G \cong \angle K$ and $\overline{FG} \cong \overline{JK}$ and $\overline{GH} \cong \overline{KL}$? No, the option $\angle G \cong \angle K$ and $\overline{FG} \cong \overline{JK}$: no, wait no, the option $\angle G \cong \angle K$ and $\overline{GH} \cong \overline{KL}$: no, that's one side. Wait wait, no, the marked angles: $\angle F \cong \angle J$, $\angle H \cong \angle L$. So for $\angle F \cong \angle J$, the sides are $\overline{FG}$ and $\overline{FH}$; $\overline{JK}$ and $\overline{JL}$. So if $\overline{FG} \cong \overline{JK}$ and $\overline{FH} \cong \overline{JL}$, that would be SAS for $\angle F \cong \angle J$. Oh right! I missed that. So $\overline{FH} \cong \overline{JL}$ and $\overline{FG} \cong \overline{JK}$ (first option) is also valid, because $\angle F \cong \angle J$ (marked), so SAS: sides around the congruent angle.
Wait no, let's list all options:
1. $\overline{FH} \cong \overline{JL}$ and $\overline{FG} \cong \overline{JK}$: $\angle F$ is between $\overline{FG}$ and $\overline{FH}$, $\angle J$ is between $\overline{JK}$ and $\overline{JL}$, and $\angle F \cong \angle J$ (marked). So this is SAS.
2. $\overline{FH} \cong \overline{JL}$ and $\overline{HG} \cong \overline{LK}$: $\angle H$ is between $\overline{FH}$ and $\overline{HG}$, $\angle L$ is between $\overline{JL}$ and $\overline{LK}$, and $\angle H \cong \angle L$ (marked). This is SAS.
3. $\angle G \cong \angle K$ and $\overline{FH} \cong \overline{JL}$: This is one angle and one non-included side, not SAS.
4. $\angle G \cong \angle K$ and $\overline{GH} \cong \overline{KL}$: This is one angle and one side, not the two sides around the angle.
5. $\angle G \cong \angle K$ and $\overline{FG} \cong \overline{JK}$: $\angle G$ is between $\overline{FG}$ and $\overline{GH}$, $\angle K$ is between $\overline{JK}$ and $\overline{KL}$. If $\angle G \cong \angle K$, $\overline{FG} \cong \overline{JK}$, but we need $\overline{GH} \cong \overline{KL}$ too. So this is only one side and angle, not SAS.

Wait no, I made a mistake earlier. The two valid options are:
- $\overline{FH} \cong \overline{JL}$ and $\overline{FG} \cong \overline{JK}$ (uses $\angle F \cong \angle J$ as the included angle)
- $\overline{FH} \cong \overline{JL}$ and $\overline{HG} \cong \overline{LK}$ (uses $\angle H \cong \angle L$ as the included angle)

Wait no, wait $\overline{FH} \cong \overline{JL}$ and $\overline{HG} \cong \overline{LK}$: $\angle H$ is between $\overline{FH}$ and $\overline{HG}$, $\angle L$ is between $\overline{JL}$ and $\overline{LK}$, yes, that's SAS.
And $\overline{FH} \cong \overline{JL}$ and $\overline{FG} \cong \overline{JK}$: $\angle F$ is between $\overline{FG}$ and $\overline{FH}$, $\angle J$ is between $\overline{JK}$ and $\overline{JL}$, yes, that's SAS.

Wait but let's check the option $\angle G \cong \angle K$ and $\overline{FG} \cong \overline{JK}$: no, that's $\angle G$ and $\overline{FG}$, but the other side around $\angle G$ is $\overline{GH}$, which would need to be congruent to $\overline{KL}$, which is not in that option. So that's not SAS.

Answer:

$\overline{FH} \cong \overline{JL}$ and $\overline{FG} \cong \overline{JK}$
$\overline{FH} \cong \overline{JL}$ and $\overline{HG} \cong \overline{LK}$