Question: Find the measures of $\angle ADB$ and $\angle BDC$ formed between the escalators.
Graph: Two intersecting diagonal escalator lines cross at point $D$ in the center. The upper-left ray goes through $A$ to the top-right ray through $B$, and the lower-left ray goes through the bottom-right ray through $C$. The angle at $D$ between $DA$ and $DB$ is labeled $(6x + 10)^\circ$ and the angle at $D$ between $DB$ and $DC$ is labeled $4x^\circ$.
1. **State the problem:** We need to find the measures of $\angle ADB$ and $\angle BDC$ given that $\angle ADB = (6x + 10)^\circ$ and $\angle BDC = 4x^\circ$.
2. **Understand the relationship:** Since $DA$, $DB$, and $DC$ are rays forming angles around point $D$, and the two angles $\angle ADB$ and $\angle BDC$ are adjacent and form a straight line, their measures add up to $180^\circ$.
3. **Write the equation:**
$$
(6x + 10) + 4x = 180
$$
4. **Simplify the equation:**
$$
6x + 10 + 4x = 180
$$
$$
10x + 10 = 180
$$
5. **Isolate $x$:**
$$
10x = 180 - 10
$$
$$
10x = 170
$$
$$
x = \frac{170}{10}
$$
$$
x = 17
$$
6. **Calculate the angles:**
$$
\angle ADB = 6x + 10 = 6(17) + 10 = 102 + 10 = 112^\circ
$$
$$
\angle BDC = 4x = 4(17) = 68^\circ
$$
7. **Check the sum:**
$$
112^\circ + 68^\circ = 180^\circ
$$
This confirms the angles are correct.
**Final answer:**
$\angle ADB = 112^\circ$
$\angle BDC = 68^\circ$