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The problem involves the angle \( \angle ACB \), which is bisected by ray \( CD \). We are given two expressions for the bisected angles:

\[
m\angle ACD = (4x + 7)^\circ \quad \text{and} \quad m\angle DCB = (7x - 38)^\circ
\]
We are tasked with determining the measure of \( \angle ACB \) and finding the value of \( x \) using an appropriate equation.

### Step 1: Concept of Angle Bisector
Since \( CD \) bisects \( \angle ACB \), the two angles on either side of the bisector must add up to \( \angle ACB \). Therefore:
\[
m\angle ACD + m\angle DCB = m\angle ACB
\]

### Step 2: Substituting the Given Expressions
Now substitute the expressions for \( m\angle ACD \) and \( m\angle DCB \) into this equation:
\[
(4x + 7) + (7x - 38) = m\angle ACB
\]
Simplify:
\[
(4x + 7x) + (7 - 38) = m\angle ACB
\]
\[
11x - 31 = m\angle ACB
\]

### Step 3: Solving for \( x \)
To find \( x \), we use the given information that \( x = 15 \), as indicated in the problem. Let's check by substituting \( x = 15 \) into the equation:
\[
11(15) - 31 = m\angle ACB
\]
\[
165 - 31 = m\angle ACB
\]
\[
m\angle ACB = 134^\circ
\]

Thus, the measure of \( \angle ACB \) is \( 134^\circ \).

### Step 4: Correct Equation for Proof
The correct equation that we would use to prove this is option **B**:
\[
m\angle ACD + m\angle DCB = m\angle ACB
\]

### Final Answer:
The correct choice is **B**.

Would you like further details or clarification on any steps? 

Here are 5 related questions:
1. How do you verify the measure of an angle using algebraic expressions?
2. What is the role of an angle bisector in geometry?
3. How would you solve for \( x \) if the measure of \( \angle ACB \) were different?
4. Can you explain the properties of supplementary and complementary angles?
5. How do you approach problems that involve systems of linear equations in geometry?

**Tip**: When solving geometric problems involving algebra, always write down what you know from the geometric relationships first before diving into solving the equations.

∠ACB is bisected by ray CD. If m∠ACD = (4x + 7)° and m∠DCB = (7x - 38)°, find the m∠ACB. Which of the following equations would you use to prove that x = 15?

Solve a geometry problem where ray CD bisects ∠ACB, and algebraic expressions are used to find m∠ACB. Learn how to solve for x using the angle bisector theorem, ideal for students in Grades 9-10.

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