Question: 14. Find m\angle VUZ if m\angle VUT = 122^\circ, m\angle VUZ = 4x + 8, and m\angle ZUT = 14x + 6.
15. If m\angle RST = 12x - 1, m\angle RSU = 9x - 15, and m\angle UST = 53^\circ, solve for x, and find m\angle RST and m\angle RSU.
1. **Problem 14:** Find $m\angle VUZ$ given $m\angle VUT = 122^\circ$, $m\angle VUZ = 4x + 8$, and $m\angle ZUT = 14x + 6$.
2. **Understanding the problem:** Angles $VUZ$, $ZUT$, and $VUT$ are adjacent angles around vertex $U$. Since $VUT$ is formed by rays $UV$ and $UT$, and $VUZ$ and $ZUT$ are parts of the angle $VUT$, we can assume that
$$m\angle VUZ + m\angle ZUT = m\angle VUT$$
3. **Write the equation:**
$$4x + 8 + 14x + 6 = 122$$
4. **Combine like terms:**
$$18x + 14 = 122$$
5. **Isolate $x$:**
$$18x = 122 - 14$$
$$18x = 108$$
6. **Divide both sides by 18:**
$$x = \frac{108}{18}$$
$$x = 6$$
7. **Find $m\angle VUZ$ by substituting $x=6$:**
$$m\angle VUZ = 4(6) + 8 = 24 + 8 = 32^\circ$$
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8. **Problem 15:** Given $m\angle RST = 12x - 1$, $m\angle RSU = 9x - 15$, and $m\angle UST = 53^\circ$, solve for $x$, then find $m\angle RST$ and $m\angle RSU$.
9. **Understanding the problem:** Angles $RSU$ and $UST$ are adjacent and together form angle $RST$. So,
$$m\angle RSU + m\angle UST = m\angle RST$$
10. **Write the equation:**
$$9x - 15 + 53 = 12x - 1$$
11. **Simplify left side:**
$$9x + 38 = 12x - 1$$
12. **Bring variables to one side and constants to the other:**
$$9x - 12x = -1 - 38$$
$$-3x = -39$$
13. **Divide both sides by -3:**
$$x = \frac{-39}{-3}$$
$$x = 13$$
14. **Find $m\angle RST$:**
$$m\angle RST = 12(13) - 1 = 156 - 1 = 155^\circ$$
15. **Find $m\angle RSU$:**
$$m\angle RSU = 9(13) - 15 = 117 - 15 = 102^\circ$$