Question: What is the value of $x$, if $\angle ADC$ is $89^\circ$?
A ray from $D$ to $A$ is on the left, a ray from $D$ to $B$ is in the middle, and a ray from $D$ to $C$ is on the right. The angle between $DA$ and $DB$ is $52^\circ$, the angle between $DB$ and $DC$ is $3x$, and the total angle $\angle ADC$ is $89^\circ$.
1. **State the problem:** We need to find the value of $x$ given that the total angle $\angle ADC$ is $89^\circ$, and it is composed of two smaller angles: $\angle ADB = 52^\circ$ and $\angle BDC = 3x$.
2. **Write the equation:** Since $\angle ADC$ is the sum of $\angle ADB$ and $\angle BDC$, we have:
$$\angle ADC = \angle ADB + \angle BDC$$
Substitute the known values:
$$89 = 52 + 3x$$
3. **Solve for $x$:**
Subtract $52$ from both sides:
$$89 - 52 = 3x$$
$$37 = 3x$$
Divide both sides by $3$:
$$x = \frac{37}{3}$$
Show cancellation:
$$x = \frac{\cancel{37}}{\cancel{3}}$$
4. **Final answer:**
$$x = 12.33$$ (approximately)