1. **State the problem:** Given that AECF is a parallelogram, triangles ABC and ADC are congruent, and segments EB and FD are congruent, prove that AECF is a rhombus. 2. **Recall definitions and properties:** - A parallelogram is a quadrilateral with opposite sides parallel. - A rhombus is a parallelogram with all sides congruent. - CPCTC (Corresponding Parts of Congruent Triangles are Congruent) allows us to conclude that corresponding angles and sides of congruent triangles are equal. 3. **Step 1:** Statement: AECF is a parallelogram, △ABC ≅ △ADC, EB ≅ FD. Reason: Given. 4. **Step 2:** Statement: ∠B ≅ ∠D. Reason: CPCTC (since △ABC ≅ △ADC). 5. **Step 3:** Statement: BC ≅ DC. Reason: CPCTC. 6. **Step 4:** Statement: AB ≅ AD. Reason: CPCTC (since △ABC ≅ △ADC, corresponding sides AB and AD are congruent). 7. **Step 5:** Statement: EC ≅ FC. Reason: CPCTC (since △ABC ≅ △ADC, corresponding segments EC and FC are congruent). 8. **Step 6:** Statement: AECF is a rhombus. Reason: AECF is a parallelogram with one pair of consecutive sides congruent (EC ≅ FC), which implies all sides are congruent, so it is a rhombus. **Final conclusion:** By showing that AECF is a parallelogram with congruent consecutive sides, we conclude AECF is a rhombus.