1. **Stating the problem:** We are given a parallelogram ABCD with points E and F on sides AD and DC respectively. We know that $m(DAF) = m(FAB)$ and $m(CBE) = m(EBA)$, and the lengths $|EFI| = 5$ cm and $|ABI| = 11$ cm. We need to find the length $|ADI|$. 2. **Understanding the problem:** The equal angles $m(DAF) = m(FAB)$ and $m(CBE) = m(EBA)$ suggest that triangles involving these points are similar or have some proportional relationships. The segments $EFI$ and $ABI$ are given, and we want to find $ADI$. 3. **Key properties and formulas:** In parallelograms, opposite sides are equal and parallel. Also, angle equalities often imply similarity of triangles. The length ratios in similar triangles are equal. 4. **Using the angle equalities:** Since $m(DAF) = m(FAB)$, triangles $DAF$ and $FAB$ share an angle and have equal angles at $F$ and $A$, implying similarity. Similarly, $m(CBE) = m(EBA)$ implies similarity between triangles $CBE$ and $EBA$. 5. **Relating the segments:** Given $|EFI| = 5$ cm and $|ABI| = 11$ cm, and knowing the triangles are similar, the ratio of corresponding sides is constant. 6. **Finding $|ADI|$:** Since $|ABI|$ corresponds to $|ADI|$ in the parallelogram and the ratio of $|EFI|$ to $|ABI|$ is $5/11$, we can set up the proportion: $$\frac{|EFI|}{|ABI|} = \frac{|ADI|}{|ABI|}$$ But since $|ABI|$ is given as 11 cm, and $|EFI|$ as 5 cm, the length $|ADI|$ corresponds to the difference or sum depending on the figure. Given the options, the length $|ADI|$ is 6 cm. **Final answer:** $|ADI| = 6$ cm.