1. **Stating the problem:** We have a triangle with angles at vertices Z, Y, and X. Given angles are $\angle Z = 47^\circ$ and $\angle Y = 40^\circ$. We need to find $\angle X$ and $\angle Y$ (though $\angle Y$ is given, so likely the problem means to confirm or find missing angles). 2. **Formula used:** The sum of interior angles in any triangle is always $180^\circ$. That is, $$\angle Z + \angle Y + \angle X = 180^\circ$$ 3. **Calculate $\angle X$:** Substitute the known angles: $$47^\circ + 40^\circ + \angle X = 180^\circ$$ 4. **Simplify:** $$\angle X = 180^\circ - 47^\circ - 40^\circ$$ 5. **Evaluate:** $$\angle X = 180^\circ - 87^\circ = 93^\circ$$ 6. **Conclusion:** - $\angle X = 93^\circ$ - $\angle Y = 40^\circ$ (given) Thus, the missing angle $\angle X$ is $93^\circ$.