Question: Find m\angle YCD if m\angle YCD = x + 58, m\angle BCY = x + 31, and m\angle BCD = 83\degree.
1. **State the problem:** We need to find the measure of angle $m\angle YCD$ given that $m\angle YCD = x + 58$, $m\angle BCY = x + 31$, and $m\angle BCD = 83\degree$.
2. **Understand the relationship:** Since $\angle BCY$ and $\angle YCD$ are adjacent angles that together form $\angle BCD$, their measures add up:
$$m\angle BCY + m\angle YCD = m\angle BCD$$
3. **Write the equation:** Substitute the given expressions:
$$(x + 31) + (x + 58) = 83$$
4. **Simplify the equation:**
$$x + 31 + x + 58 = 83$$
$$2x + 89 = 83$$
5. **Isolate $x$:**
$$2x = 83 - 89$$
$$2x = -6$$
6. **Solve for $x$:**
$$x = \frac{-6}{2}$$
$$x = \cancel{\frac{\textcolor{red}{-6}}{\textcolor{red}{2}}} = -3$$
7. **Find $m\angle YCD$ by substituting $x$ back:**
$$m\angle YCD = x + 58 = -3 + 58 = 55\degree$$
**Final answer:**
$$m\angle YCD = 55\degree$$