The given information about triangle congruence and the midpoint of a side allows us to apply the AAS congruence criterion to prove that the two triangles are congruent.
Given:
Triangle BAC is congruent to triangle DEC.
C is the midpoint of AE.
Prove:
Triangle AABC is congruent to triangle AEDC.
Proof:
HL Congruency: Since C is the midpoint of AE, then AC = CE. This is given in the problem statement.
Angle Congruency: We are also given that BAC = DEC.
AAS Congruency: Together, statements 1 and 2 satisfy the Angle-Angle-Side (AAS) congruence criterion. Therefore, triangle AABC is congruent to triangle AEDC.
Detailed Explanation of the Proof:
Step 1: Understanding the AAS Congruence Criterion:
The AAS congruence criterion states that if two triangles have two pairs of corresponding angles and a non-included side (the side not shared by the angles) equal, then the triangles are congruent. This means that the triangles have the same size and shape.
Step 2: Applying AAS to the Given Triangles:
Angle Congruency: We are given that BAC ≅ DEC from the problem statement. This satisfies the first part of the AAS criterion.
Side Congruency: Since C is the midpoint of AE, then AC = CE (by definition of midpoint). This fulfills the requirement for the non-included side in the AAS criterion.
Angle Congruency: We return to the given information and note that ∠BCA ≅ ∠DCE due to their shared side BC and the angle sum property of triangles (180 degrees). This completes the second pair of congruent angles in the AAS criterion.
Step 3: Conclusion and Congruence:
Having satisfied all three conditions of the AAS criterion, we can confidently conclude that triangle AABC ≅ AEDC. This means that the corresponding sides and angles of the two triangles are equal in measure, proving that they are congruent.
Additional Information:
The AAS criterion is similar to the ASA (Angle-Side-Angle) congruence criterion, but it uses a non-included side instead of the included side between the angles.
Understanding the logic behind the AAS and other congruence criteria can help you solve various geometrical problems involving triangles and their properties.