1. **Stating the problem:** We have a triangle with three internal angles labeled as $z^\circ$, $y^\circ$, and $x^\circ$. Given are some external angles and one internal angle $X = 113^\circ$. We need to find the values of $x$, $y$, and $z$. 2. **Recall the triangle angle sum rule:** The sum of internal angles in any triangle is always $$x + y + z = 180^\circ.$$ 3. **Use the external angle property:** An external angle is equal to the sum of the two opposite internal angles. For example, if an external angle adjacent to angle $z$ is $28^\circ$, then $$28^\circ = x + y.$$ 4. **Given angle $X = 113^\circ$ is an internal angle:** Since $X$ corresponds to $x$, we have $$x = 113^\circ.$$ 5. **Use the external angle $28^\circ$ adjacent to $z$:** $$28^\circ = x + y = 113^\circ + y.$$ 6. **Solve for $y$:** $$y = 28^\circ - 113^\circ = -85^\circ.$$ This is impossible for an angle in a triangle, so re-examine the external angle assignment. 7. **Check the other external angle $39^\circ$ adjacent to $y$ or $x$:** If $39^\circ$ is external to angle $y$, then $$39^\circ = x + z.$$ 8. **Substitute $x = 113^\circ$:** $$39^\circ = 113^\circ + z \implies z = 39^\circ - 113^\circ = -74^\circ,$$ which is also impossible. 9. **Reconsider the problem setup:** The internal angles are $64^\circ$, $28^\circ$, and $39^\circ$ as given, so these must be the values of $x$, $y$, and $z$. 10. **Sum check:** $$64^\circ + 28^\circ + 39^\circ = 131^\circ,$$ which is less than $180^\circ$, so these cannot be all internal angles. 11. **Use the fact that external angle + internal angle at a vertex = $180^\circ$:** - For angle $x$ with external angle $113^\circ$, internal angle is $$x = 180^\circ - 113^\circ = 67^\circ.$$ - For external angle $28^\circ$ adjacent to $z$, internal angle is $$z = 180^\circ - 28^\circ = 152^\circ.$$ - For external angle $39^\circ$ adjacent to $y$, internal angle is $$y = 180^\circ - 39^\circ = 141^\circ.$$ 12. **Sum check:** $$x + y + z = 67^\circ + 141^\circ + 152^\circ = 360^\circ,$$ which is impossible for a triangle. 13. **Conclusion:** The problem statement is ambiguous or incomplete. However, the only consistent internal angle given is $x = 113^\circ$. **Final answer:** $$x = 113^\circ.$$