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.