1. **State the problem:** We need to determine which congruence test is used to prove that the purple triangle is congruent to the red triangle given the information about angles and sides. 2. **Recall the congruence tests:** The common triangle congruence tests are: - SSS (Side-Side-Side): All three sides equal. - SAS (Side-Angle-Side): Two sides and the included angle equal. - ASA (Angle-Side-Angle): Two angles and the included side equal. - AAS (Angle-Angle-Side): Two angles and a non-included side equal. 3. **Analyze the given information:** - Step 1: $a = y$ (interior alternate angles are equal) - Step 2: $b = x$ (interior alternate angles are equal) - Step 3: $EF = IJ$ (given side equality) 4. **Identify the order of elements:** The two equal angles $a$ and $b$ correspond to the two angles in each triangle, and the side $EF = IJ$ is between these two angles. 5. **Apply the ASA test:** ASA requires two angles and the included side to be equal. Here, the side $EF = IJ$ is between angles $a$ and $b$, so the triangles are congruent by ASA. 6. **Conclusion:** The congruence test used is **ASA (Angle-Side-Angle)**. **Final answer:** The triangles are congruent by the ASA congruence test.