Question

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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°
□ $overline{eg} cong overline{eg}$
image of triangles efg (angles at f: 66°, e: 72°) and efg (angles at f: 66°, e: 72°)

Explanation:

First, recall the AAS (Angle - Angle - Side) congruence criterion for triangles: 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 two triangles are congruent.

In $\triangle EFG$ and $\triangle E'F'G'$:

• We know that $\angle F = 66^{\circ},\angle E=72^{\circ}$ and in $\triangle E'F'G'$, $\angle F' = 66^{\circ},\angle E' = 72^{\circ}$. So, $\angle F\cong\angle F'$ and $\angle E\cong\angle E'$.

Now, let's analyze each option:

Option 1: $EG = 12$ and $E'G'=12$

• In $\triangle EFG$, side $EG$ is opposite $\angle F$ (since $\angle F = 66^{\circ}$, and in a triangle, the side opposite an angle is the side that does not form the angle). In $\triangle E'F'G'$, side $E'G'$ is opposite $\angle F'$ (since $\angle F'=66^{\circ}$). Since $\angle F\cong\angle F'$, $\angle E\cong\angle E'$ and $EG\cong E'G'$ (as $EG = E'G'=12$), by AAS, $\triangle EFG\cong\triangle E'F'G'$. So this option is valid.

Option 2: $FG = 15$ and $F'G' = 15$

• In $\triangle EFG$, side $FG$ is opposite $\angle E$ (since $\angle E = 72^{\circ}$). In $\triangle E'F'G'$, side $F'G'$ is opposite $\angle E'$ (since $\angle E'=72^{\circ}$). Since $\angle E\cong\angle E'$, $\angle F\cong\angle F'$ and $FG\cong F'G'$ (as $FG = F'G' = 15$), by AAS, $\triangle EFG\cong\triangle E'F'G'$. So this option is valid.

Option 3: $EF = 10$ and $E'F'=12$

• If $EF = 10$ and $E'F'=12$, then $EF

ot\cong E'F'$. Also, $EF$ is a side included between $\angle E$ and $\angle F$, and $E'F'$ is included between $\angle E'$ and $\angle F'$. This does not satisfy AAS (we need a non - included side to be congruent), so this option is invalid.

Option 4: $m\angle G=42^{\circ}$ and $m\angle G' = 42^{\circ}$

• This just gives us another pair of congruent angles. But AAS requires two angles and a non - included side. We already know two pairs of angles are congruent, but we don't have a side congruence here. So this option does not help us prove congruence by AAS (it just gives more angle information, not the side needed for AAS). So this option is invalid.

Option 5: $\overline{EG}\cong\overline{E'G'}$

• As we analyzed in Option 1, $EG$ is opposite $\angle F$ and $E'G'$ is opposite $\angle F'$. Since $\angle F\cong\angle F'$, $\angle E\cong\angle E'$ and $\overline{EG}\cong\overline{E'G'}$, by AAS, $\triangle EFG\cong\triangle E'F'G'$. So this option is valid.

Answer:

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