Question

what additional information could be used to prove \\( \triangle efg \cong \triangle efg \\) using aas? check all that apply \\( \square eg = 12 \\) and \\( eg = 12 \\) \\( \square fg = 15 \\) and \\( fg = 15 \\) \\( \square ef = 10 \\) and \\( ef = 12 \\) \\( \square m\angle g = 42^{\circ} \\) and \\( m\angle g = 42^{\circ} \\) \\( \square \overline{eg} \cong \overline{eg} \\)

Explanation:

Step1: Recall AAS (Angle - Angle - Side) criterion

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

Step2: Analyze each option

• For \(EG = 12\) and \(E'G'=12\) (\(\overline{EG}\cong\overline{E'G'}\)):

We already have \(\angle F=\angle F' = 66^{\circ}\) and \(\angle E=\angle E'=72^{\circ}\). If \(EG = E'G'\) (a non - included side for the given angles), by AAS \(\triangle EFG\cong\triangle E'F'G'\).

• For \(FG = 15\) and \(F'G' = 15\):

The side \(FG\) (or \(F'G'\)) is included between \(\angle F\) (or \(\angle F'\)) and \(\angle G\) (or \(\angle G'\)), so it does not satisfy the non - included side requirement of AAS.

• For \(EF = 10\) and \(E'F'=12\):

Since \(EF
eq E'F'\), this does not help in proving congruence.

• For \(m\angle G = 42^{\circ}\) and \(m\angle G'=42^{\circ}\):

We have \(\angle F=\angle F' = 66^{\circ}\), \(\angle E=\angle E' = 72^{\circ}\), and now \(\angle G=\angle G'=42^{\circ}\). Also, if we consider the non - included side (for example, if we take the side opposite to the \(42^{\circ}\) angle). In terms of AAS (we can use the two angles (\(\angle E=\angle E'\), \(\angle G=\angle G'\)) and a non - included side (say \(EF\) and \(E'F'\) or \(EG\) and \(E'G'\))). But if we just consider the angle part, along with a non - included side (which can be \(EG\) and \(E'G'\) as we saw above).

• For \(\overline{EG}\cong\overline{E'G'}\):

Combined with \(\angle F=\angle F'\) and \(\angle E=\angle E'\), by AAS \(\triangle EFG\cong\triangle E'F'G'\)

Answer:

• EG = 12 and E'G' = 12
• m∠G = 42° and m∠G' = 42°
• \(\overline{EG}\cong\overline{E'G'}\)