Final answer:
Using the Side-Angle-Side (SAS) Postulate, we prove that triangles ABC and BAD are congruent because segments AC and BD are congruent, angles CAB and DBA are congruent, and side AB is congruent to itself.
Step-by-step explanation:
To prove that triangles ABC and BAD are congruent given that segment —AC is congruent to segment BD and angle ∠CAB is congruent to angle ∠DBA, we can use the SAS (Side-Angle-Side) Postulate. The SAS Postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
Step 1: Identify the two sides and the included angle in each triangle. For triangle ABC, the two sides are AC and AB, and the included angle is ∠CAB. For triangle BAD, the two sides are BD and BA, and the included angle is ∠DBA.
Step 2: Use the given information to establish congruence. Since segment —AC is congruent to segment BD, and angle ∠CAB is congruent to angle ∠DBA, we only need to prove that the third side of each triangle is congruent to establish that the triangles are congruent by SAS.
Step 3: Show that the third side of each triangle is congruent. Since segment AB is shared by both triangles ABC and BAD, it is congruent to itself by the Reflexive Property of Congruence.
Therefore, by the SAS Postulate, we have proven that triangle ABC is congruent to triangle BAD.