1. **Stating the problem:** We are given a circle with points E, S, T and angles related to these points. We need to find the measures of angles and segments: $m\angle E$, $mSET$, $m\angle I$, $mIS$, $m\angle IST$, $m\angle T$, and $m\angle ESI + m\angle ETI$. 2. **Given information:** - $mSET = 2m\angle I = 180^\circ$ - $m\angle I = 35^\circ$ 3. **Check the calculation for $mSET$:** - Since $mSET = 2m\angle I$, then $mSET = 2 \times 35^\circ = 70^\circ$ - The user's calculation $2(145) = 290$ is incorrect because $m\angle I$ is $35^\circ$, not $145^\circ$. 4. **Find $mSET$ correctly:** $$mSET = 2 \times 35^\circ = 70^\circ$$ 5. **Find $m\angle E$:** - If $mSET = 70^\circ$, and $mSET$ is an arc measure, then $m\angle E$ (an inscribed angle intercepting the same arc) is half of $mSET$: $$m\angle E = \frac{mSET}{2} = \frac{70^\circ}{2} = 35^\circ$$ 6. **Find $mIS$ and $m\angle I$:** - Given $m\angle I = 35^\circ$ (already known). - $mIS$ is a segment length, which cannot be determined from angle measures alone without additional data. 7. **Find $m\angle IST$ and $m\angle T$:** - Without additional information or figure details, these cannot be determined precisely. 8. **Find $m\angle ESI + m\angle ETI$:** - Without specific angle measures or relationships, this sum cannot be calculated. **Summary:** The user's calculation of $mSET$ as 290 is incorrect. The correct $mSET$ is $70^\circ$ based on $m\angle I = 35^\circ$. Other measures require more information. **Final answer:** $$mSET = 70^\circ$$ $$m\angle E = 35^\circ$$ $$m\angle I = 35^\circ$$ Other values cannot be determined with given data.