Question

using angle-angle-side congruence theorem
what additional information could be used to prove
δefg ≅ δefg using aas? check all that apply.
□ eg = 12 and eg = 12
□ fg = 15 and fg = 15
□ ef = 10 and ef = 12
□ m∠g = 42° and m∠g = 42°
□ eg ≅ eg
(image of two triangles δefg and δefg with angles labeled: ∠f = 66°, ∠e = 72° in δefg; ∠f = 66°, ∠e = 72° in δefg)

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.

First, let's find the third angle of each triangle. In $\triangle EFG$, we know $\angle E = 72^{\circ}$ and $\angle F=66^{\circ}$. The sum of angles in a triangle is $180^{\circ}$, so $\angle G=180-(72 + 66)=42^{\circ}$. In $\triangle E'F'G'$, $\angle E' = 72^{\circ}$ and $\angle F'=66^{\circ}$, so $\angle G'=180-(72 + 66) = 42^{\circ}$. So we already know that $\angle E\cong\angle E'$, $\angle F\cong\angle F'$ and $\angle G\cong\angle G'$.

Step2: Analyze each option

• Option 1: $EG = 12$ and $E'G'=12$. $EG$ is a non - included side between $\angle E$ and $\angle G$, and $E'G'$ is the corresponding non - included side between $\angle E'$ and $\angle G'$. Since we have two angles ($\angle E\cong\angle E'$, $\angle G\cong\angle G'$) and a non - included side ($EG\cong E'G'$), this satisfies AAS.
• Option 2: $FG = 15$ and $F'G'=15$. $FG$ is a non - included side between $\angle F$ and $\angle G$, and $F'G'$ is the corresponding non - included side between $\angle F'$ and $\angle G'$. We have $\angle F\cong\angle F'$, $\angle G\cong\angle G'$ and $FG\cong F'G'$, so this satisfies AAS.
• Option 3: $EF = 10$ and $E'F'=12$. $EF$ and $E'F'$ are not equal, so they are not congruent. This does not satisfy AAS.
• Option 4: $m\angle G = 42^{\circ}$ and $m\angle G'=42^{\circ}$. We already know that $\angle G\cong\angle G'$ from the angle - sum property. This does not give us a new side or a new angle - side relationship for AAS (we already have the angle information).
• Option 5: $\overline{EG}\cong\overline{E'G'}$. This is the same as option 1, just written in congruence notation. It satisfies AAS as we have two angles and a non - included side.

Answer:

The correct options are:

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