To determine the length of side \( a \) in the right triangle, we first need to clarify the congruence information provided.
According to the AAS (Angle-Angle-Side) Congruence Theorem, if two angles and the 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. Since it is stated that the triangles are congruent by the AAS Congruence Theorem, this means that two angles and one corresponding side from the first triangle match the corresponding elements of the second triangle.
From the left triangle, we have:
- Angles: 70 degrees, 50 degrees, and 60 degrees
- Sides labeled as (7), (5), and (12).
If we assume that side \( a \) corresponds to one of the sides from the left triangle that is marked with tick marks, we would need to find out which of the sides (7, 5, or 12) is congruent to \( a \).
If we make the assumption that the angle configuration in both triangles fits the AAS criterion (with sides and angles properly matched), we can infer:
- If \( a \) corresponds to side (7), then \( a = 7 \).
- If \( a \) corresponds to side (5), then \( a = 5 \).
- If \( a \) corresponds to side (12), then \( a = 12 \).
Without more specific information on which side \( a \) directly corresponds to, we cannot definitively give the length of \( a \).
You would need to check which side in the right triangle is marked with tick marks congruent to the side of the left triangle to conclude which length \( a \) corresponds to.
For the sake of answering the question, let’s assume \( a \) corresponds to one of the sides with tick marks say side (7).
Thus, according to the AAS Congruence Theorem, side \( a \) must be 7.