Triangles ABC and CDA are similar by the Side-Angle-Side (SAS) criterion. Given AB ≈ CD, BC ≈ AD, and angle BCA = angle DAC, the triangles share proportional sides and congruent angles.
The given proposition involves proving the similarity of triangles ABC and CDA based on the Side-Angle-Side (SAS) similarity criterion. Initially, it's given that AB is approximately equal to CD, and BC is approximately equal to AD. These equalities establish the basis for comparing the corresponding sides of the two triangles.
To employ the SAS similarity criterion, it's crucial to establish that the included angles are congruent. In this case, angle BCA and angle DAC are identified as congruent due to the vertical angle property. Vertical angles formed by the intersection of two lines are always equal.
Therefore, with AB ≈ CD, BC ≈ AD, and angle BCA = angle DAC, the SAS condition is satisfied. The SAS similarity criterion states that if two sides of one triangle are proportional to two sides of another triangle, and their included angles are congruent, then the triangles are similar.
In conclusion, triangles ABC and CDA are similar according to the SAS criterion. This proof demonstrates a systematic approach to establishing triangle similarity based on corresponding sides and included angles, providing a solid foundation for geometric analysis.