Question: 14. In the figure, CB and CD are opposite rays, CE bisects $\angle DCF$, and CG bisects $\angle FCB$.
15. If $m\angle DCE = 4x + 15$ and $m\angle ECF = 6x - 5$, find $m\angle DCE$.
16. If $m\angle FCG = 9x + 3$ and $m\angle GCB = 13x - 9$, find $m\angle GCB$.
17. TRAFFIC SIGNS The diagram shows a sign used to warn drivers of a school zone or crossing. Measure and classify each numbered angle.
1. **Problem Statement:** Given that CB and CD are opposite rays, CE bisects $\angle DCF$, and CG bisects $\angle FCB$. We are asked to find $m\angle DCE$ given $m\angle DCE = 4x + 15$ and $m\angle ECF = 6x - 5$.
2. **Understanding the problem:** Since CE bisects $\angle DCF$, it divides $\angle DCF$ into two equal angles: $\angle DCE$ and $\angle ECF$. Therefore, $m\angle DCE = m\angle ECF$.
3. **Set up the equation:**
$$4x + 15 = 6x - 5$$
4. **Solve for $x$:**
$$4x + 15 = 6x - 5$$
$$15 + 5 = 6x - 4x$$
$$20 = 2x$$
$$x = \frac{20}{2}$$
$$x = 10$$
5. **Find $m\angle DCE$ by substituting $x=10$ into $4x + 15$:**
$$m\angle DCE = 4(10) + 15 = 40 + 15 = 55$$
6. **Answer:**
$$m\angle DCE = 55^\circ$$