https://file.aiquickdraw.com/mathbot/file/17283035435861u32mutv.jpg

The problem in the image states:

**Given:**
- \( \overrightarrow{BD} \) bisects \( \angle ABC \), meaning that \( \angle DBC = \angle ABD \).
- \( m\angle DBC = x + 2 \)
- \( m\angle ABD = 2x - 5 \)

**To find:**
The value of \( x \).

### Solution:

Since \( \overrightarrow{BD} \) bisects \( \angle ABC \), the two angles \( \angle DBC \) and \( \angle ABD \) must be equal. Therefore, we can set up the equation:

\[
x + 2 = 2x - 5
\]

Now, solve for \( x \):

1. Subtract \( x \) from both sides:
   \[
   2 = x - 5
   \]

2. Add 5 to both sides:
   \[
   7 = x
   \]

Thus, the value of \( x \) is \( \boxed{7} \).

### Final Answer:
The value of \( x \) is \( 7 \), corresponding to option **C**.

Would you like further clarification on any step or more details?

Here are some related questions to explore:

1. How does angle bisector theorem apply in other geometric problems?
2. What are the properties of bisected angles in different polygons?
3. Can this technique be used with supplementary angles?
4. How do you handle cases where bisected angles have expressions involving multiple variables?
5. What is the relationship between bisectors and symmetry in triangles?

**Tip:** Always double-check if the angles are equal when you have a bisector in a geometric problem, as it simplifies the solving process.

BD bisects ∠ABC so that m∠DBC = x + 2 and m∠ABD = 2x - 5. What is the value of x?

Learn how to solve a problem involving the bisector of an angle ∠ABC, where m∠DBC = x + 2 and m∠ABD = 2x - 5. This problem includes the application of the Angle Bisector Theorem and algebraic equation solving. Suitable for students in Grades 8-10.

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