1. **Stating the problem:** We need to find the measure of angle $HFE$ in the polygon with vertices $H, E, G, F$. Given are sides $HE=38$ cm, $EG=35$ cm, $GF=60$ cm, angles $\angle HEG=62^\circ$, $\angle EGF=107^\circ$, and a right angle between $HE$ and $EF$. The triangle $HEF$ is highlighted. 2. **Understanding the problem:** Since $HE$ and $EF$ are perpendicular, $\angle HEF=90^\circ$. We want to find $\angle HFE$ in triangle $HEF$. 3. **Using triangle angle sum rule:** In triangle $HEF$, the sum of angles is $180^\circ$: $$\angle HEF + \angle HFE + \angle EHF = 180^\circ$$ We know $\angle HEF=90^\circ$, so: $$90^\circ + \angle HFE + \angle EHF = 180^\circ$$ which simplifies to: $$\angle HFE + \angle EHF = 90^\circ$$ 4. **Finding $\angle EHF$:** $\angle HEG=62^\circ$ is the angle at $E$ between $H$ and $G$. Since $HE$ and $EF$ are perpendicular, $\angle HEF=90^\circ$, and $EF$ lies along a line perpendicular to $HE$. The angle $\angle EHF$ is supplementary to $\angle HEG$ because $F$ lies on the line perpendicular to $HE$ at $E$. Therefore: $$\angle EHF = 180^\circ - 62^\circ = 118^\circ$$ 5. **Calculate $\angle HFE$:** Using step 3: $$\angle HFE = 90^\circ - \angle EHF = 90^\circ - 118^\circ = -28^\circ$$ This negative value indicates a misinterpretation; instead, $\angle EHF$ is inside triangle $HEF$, so it cannot be $118^\circ$. Let's reconsider. 6. **Reconsidering $\angle EHF$:** Since $HE$ and $EF$ are perpendicular, $\angle HEF=90^\circ$. The angle $\angle HEG=62^\circ$ is at $E$ between $H$ and $G$. The angle $\angle EHF$ is the angle at $H$ between $E$ and $F$. We need to find $\angle EHF$. 7. **Using Law of Cosines in triangle $HEF$:** We know $HE=38$ cm, $EF$ is unknown, and $HF$ is unknown. We need more information to find $\angle HFE$. 8. **Using the polygon angles:** The sum of interior angles in quadrilateral $HEGF$ is $360^\circ$. Given $\angle HEG=62^\circ$, $\angle EGF=107^\circ$, and $\angle GFE$ unknown, and $\angle FHE$ unknown, but this is complex without more data. 9. **Conclusion:** With the given data and the right angle at $HEF$, the angle $HFE$ in triangle $HEF$ is complementary to $\angle EHF$: $$\angle HFE = 90^\circ - \angle EHF$$ Since $\angle EHF$ is not given, it cannot be determined exactly from the provided information. **Final answer:** The measure of angle $HFE$ depends on $\angle EHF$ and satisfies: $$\boxed{\angle HFE = 90^\circ - \angle EHF}$$