Question

using angle-angle-side congruence theorem
what additional information could be used to prove \\(\triangle efg \cong \triangle efg\\) using aas?
choose three correct answers.
\\(ef = 10\\) and \\(ef = 12\\)
\\(\overline{eg} \cong \overline{eg}\\)
\\(eg = 12\\) and \\(eg = 12\\)
\\(m\angle 42^{\circ}\\) and \\(m\angle g = 42^{\circ}\\)
\\(fg = 15\\) and \\(fg = 15\\)

Explanation:

Step1: Recall AAS Congruence

AAS (Angle - Angle - Side) congruence states that if two angles and a non - included side of one triangle are congruent to the corresponding two angles and non - included side of another triangle, then the triangles are congruent.

In $\triangle EFG$ and $\triangle E'F'G'$, we know that $\angle F=\angle F' = 66^{\circ}$ and $\angle E=\angle E'=72^{\circ}$. So we need a non - included side to be congruent.

Step2: Analyze each option

• Option 1: $EF = 10$ and $E'F'=12$: This gives information about included sides (since $\angle F$ and $\angle E$ are the angles, the included side is $EF$), so this is not for AAS.
• Option 2: $\overline{EG}\cong\overline{E'G'}$: $\angle F=\angle F'$ and $\angle E=\angle E'$, and $EG$ and $E'G'$ are non - included sides (the side opposite $\angle F$ and $\angle F'$ respectively). So this can be used for AAS.
• Option 3: $FG = 15$ and $F'G'=15$: $\angle E=\angle E'$ and $\angle F=\angle F'$, and $FG$ and $F'G'$ are non - included sides (the side opposite $\angle E$ and $\angle E'$ respectively). So this can be used for AAS.
• Option 4: $EG = 12$ and $E'G'=12$: $\angle F=\angle F'$ and $\angle E=\angle E'$, and $EG$ and $E'G'$ are non - included sides (the side opposite $\angle F$ and $\angle F'$ respectively). So this can be used for AAS.
• Option 5: $m\angle G = 42^{\circ}$ and $m\angle G'=42^{\circ}$: We already know two angles of each triangle, and the sum of angles in a triangle is $180^{\circ}$, so we can find the third angle. But this is just confirming the third angle, not providing a side for AAS.

Answer:

The three correct options are:

• $\overline{EG}\cong\overline{E'G'}$
• $FG = 15$ and $F'G'=15$
• $EG = 12$ and $E'G'=12$