1. The problem states that $D \perp C$ and $A \perp B$, indicating right angles at these points. 2. We are asked to identify the postulate that makes the triangles congruent. 3. The right angles give us one pair of congruent angles. 4. Since $E$ lies on the line between $D$ and $B$, and the triangles share side $DE$ or $EB$, we have a pair of congruent sides. 5. If another pair of angles or sides is congruent, we can use the ASA (Angle-Side-Angle) or SAS (Side-Angle-Side) postulate. 6. Given the right angles and shared side, and assuming the other angles or sides are congruent, the ASA postulate applies. 7. Therefore, the triangles are congruent by the ASA postulate. Final answer: ASA