Question

2.
△efg ~ △_ehg_
by ______

Explanation:

Step1: Identify Common Angles

From the diagram, $\angle FEG = \angle HEG$ (marked angles) and $\angle FGE = \angle HGE$ (marked angles). Also, $EG$ is common to both $\triangle EFG$ and $\triangle EHG$.

Step2: Determine Similarity Criterion

Using the Angle - Angle (AA) similarity criterion (if two angles of one triangle are equal to two angles of another triangle, the triangles are similar) or by Side - Angle - Side (SAS) if we consider the included angle. But from the marked angles, $\angle FEG=\angle HEG$, $EG = EG$, and $\angle FGE=\angle HGE$, so by ASA (Angle - Side - Angle) or AA, the triangles $\triangle EFG$ and $\triangle EHG$ are similar. Wait, actually, let's re - check. The common side is $EG$, and we have two pairs of equal angles. So $\triangle EFG \sim \triangle EHG$ by the AA (Angle - Angle) similarity criterion (since two angles are equal, the third must be equal too) or by ASA. But more precisely, looking at the angles: $\angle FEG=\angle HEG$ (let's call this $\angle1$), $\angle FGE=\angle HGE$ (let's call this $\angle2$), and $EG$ is common. So by ASA (Angle - Side - Angle) congruence? Wait, the problem says $\sim$ (similar) or $\cong$ (congruent)? The notation is $\triangle EFG \sim \triangle \_\_\_$. So let's see the angles. $\angle FEG$ and $\angle HEG$ are equal, $\angle FGE$ and $\angle HGE$ are equal, so $\triangle EFG \sim \triangle EHG$ by AA (Angle - Angle) similarity (because if two angles of one triangle are equal to two angles of another triangle, the triangles are similar).

Answer:

$\triangle EFG \sim \triangle \boldsymbol{EHG}$ by AA (Angle - Angle) Similarity Criterion (or ASA, but AA is more appropriate for similarity)