Given that $\overrightarrow{BD}$ bisects $\angle ABC$, it means that $\angle ABD$ and $\angle DBC$ are equal. This gives us the equation:

$\angle ABD = \angle DBC$

From the image:

• $\angle ABD = (3x - 7)^\circ$
• $\angle DBC = 20^\circ$

Since $\overrightarrow{BD}$ bisects $\angle ABC$, we can set up the equation:

$3x - 7 = 20$

Now, let's solve for $x$:

1. Add 7 to both sides of the equation:

$3x = 27$

2. Divide by 3:

$x = 9$

So, the value of $x$ is $9$.

Would you like more details or have any questions about this solution?

Here are 5 related questions that build on this information:

1. What is the measure of $\angle ABC$ in the diagram?
2. How do you find the measure of an angle when given an algebraic expression?
3. How can you check if a line truly bisects an angle?
4. What is the importance of angle bisectors in triangle properties?
5. How can this concept of angle bisectors be applied in real-life scenarios?

Tip: In geometry, when a line bisects an angle, it divides the angle into two equal parts, which is a helpful property for solving various problems.