which rule explains why these triangles are congruent? sss aas asa sas these triangles cannot be proven congruent.

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# Brief Explanations:
1. First, identify the vertical angles at $ V $: $ \angle XVW $ and $ \angle ZVY $ are vertical angles, so they are equal.
2. Then, look at the marked angles: $ \angle X \cong \angle Y $ (double arcs) and $ \angle W \cong \angle Z $ (single arcs). Wait, no, actually, let's re - examine. Wait, the triangles are $ \triangle XVW $ and $ \triangle YVZ $? Wait, no, the vertices: $ X, V, W $ and $ Z, V, Y $. The angles: $ \angle X $ and $ \angle Y $ (double arcs), $ \angle W $ and $ \angle Z $ (single arcs), and the vertical angles at $ V $ ( $ \angle XVW=\angle ZVY $ ). So we have two angles and a non - included side? Wait, no, AAS (Angle - Angle - Side) 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, the triangles are congruent. Let's check:
   - $ \angle X\cong\angle Y $ (given by the double arcs).
   - $ \angle XVW\cong\angle ZVY $ (vertical angles).
   - The side between? Wait, no, AAS: two angles and a side that is not between them. Wait, actually, in $ \triangle XVW $ and $ \triangle YVZ $, we have $ \angle X\cong\angle Y $, $ \angle XVW\cong\angle ZVY $, and the side $ VW $ and $ VZ $? No, wait, the vertical angles are at $ V $, so the sides: the side opposite to one of the angles. Wait, maybe I made a mistake. Wait, let's list the congruent parts:
     - $ \angle X=\angle Y $ (marked with two arcs).
     - $ \angle W=\angle Z $ (marked with one arc).
     - The vertical angles $ \angle XVW=\angle ZVY $. Wait, no, actually, the triangles are $ \triangle XW V $ and $ \triangle ZY V $. Wait, the vertical angles are $ \angle XVW $ and $ \angle ZVY $. So we have two angles ( $ \angle X $ and $ \angle W $ in $ \triangle XVW $, and $ \angle Y $ and $ \angle Z $ in $ \triangle ZVY $) and the included side? No, AAS: if two angles and a non - included side are congruent. Wait, let's recall the AAS criterion: AAS (Angle - Angle - Side) means that two angles and a side that is not between them are congruent. In this case, we have $ \angle X\cong\angle Y $, $ \angle XVW\cong\angle ZVY $ (vertical angles), and the side $ VW $ and $ VZ $? No, wait, the side opposite to one of the angles. Wait, actually, the correct way: in $ \triangle XVW $ and $ \triangle YVZ $, we have:
       - $ \angle X\cong\angle Y $ (given by the markings).
       - $ \angle XVW\cong\angle ZVY $ (vertical angles, so congruent).
       - The side $ VW $ and $ VZ $? No, wait, the side between $ \angle X $ and $ \angle XVW $ is $ XV $, and between $ \angle Y $ and $ \angle ZVY $ is $ YV $? No, maybe I messed up the triangles. Wait, the triangles are $ \triangle XWV $ and $ \triangle ZYV $. Wait, the angles: $ \angle X $ and $ \angle Y $ (double arcs), $ \angle W $ and $ \angle Z $ (single arcs), and the vertical angles at $ V $. So we have two angles and a side? Wait, AAS: two angles and a non - included side. So if we have two angles (e.g., $ \angle X $ and $ \angle W $) and the side $ XW $, but no, the vertical angles are at $ V $. Wait, actually, the correct congruence rule here is AAS. Because we have two angles ( $ \angle X\cong\angle Y $, $ \angle W\cong\angle Z $) and the side $ VW \cong VZ $? No, wait, the vertical angles are equal, and two other angles. Wait, let's start over.
     - Vertical angles: $ \angle XVW=\angle ZVY $ (vertical angles are congruent).
     - Given $ \angle X=\angle Y $ (double arcs) and $ \angle W=\angle Z $ (single arcs). Wait, no, the markings: $ \angle X $ and $ \angle Y $ have double arcs, $ \angle W $ and $ \angle Z $ have single arcs. So in $ \triangle XVW $ and $ \triangle YVZ $:
       - $ \angle X\cong\angle Y $ (given).
       - $ \angle XVW\cong\angle ZVY $ (vertical angles).
       - The side $ VW $ and $ VZ $? No, the side opposite to $ \angle X $ in $ \triangle XVW $ is $ VW $, and opposite to $ \angle Y $ in $ \triangle YVZ $ is $ VZ $? No, maybe the triangles are $ \triangle XWV $ and $ \triangle ZYV $. Wait, the key is that we have two angles and a non - included side. So AAS: Angle - Angle - Side. So the correct rule is AAS.
# Answer:
B. AAS

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