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We are given the following angle measures and relationships in the diagram:

1. \( m\angle ABD = 96^\circ \)
2. \( m\angle CBD = 6x + 27^\circ \)
3. \( m\angle ABC = 7x - 9^\circ \)

From the diagram, we see that:

- \( \angle ABD \) is the entire angle formed by \( \angle ABC \) and \( \angle CBD \). This means:
\[
m\angle ABD = m\angle ABC + m\angle CBD
\]

### Step 1: Set up the equation
Using the given information, we substitute the expressions for \( m\angle ABC \) and \( m\angle CBD \) into the equation:

\[
96^\circ = (7x - 9^\circ) + (6x + 27^\circ)
\]

### Step 2: Solve for \( x \)
Simplify the equation:

\[
96^\circ = 7x - 9^\circ + 6x + 27^\circ
\]
\[
96^\circ = 13x + 18^\circ
\]

Now, subtract 18° from both sides:

\[
96^\circ - 18^\circ = 13x
\]
\[
78^\circ = 13x
\]

Now, divide by 13 to solve for \( x \):

\[
x = \frac{78^\circ}{13} = 6
\]

### Step 3: Find \( m\angle CBD \)
Now that we know \( x = 6 \), substitute it into the expression for \( m\angle CBD \):

\[
m\angle CBD = 6x + 27^\circ = 6(6) + 27^\circ = 36^\circ + 27^\circ = 63^\circ
\]

Thus, the measure of \( \angle CBD \) is \( 63^\circ \).

---

Would you like a detailed explanation of any step, or do you have any questions? Here are some related problems you might find interesting:

1. What is the value of \( m\angle ABC \)?
2. How would the result change if \( m\angle ABD \) were increased by 10°?
3. Can you solve for \( m\angle ABC \) in terms of \( m\angle ABD \) without the diagram?
4. What is the relationship between \( m\angle ABC \) and \( m\angle CBD \) if \( m\angle ABD \) is 120°?
5. What happens to the angles if \( x \) is a negative value?

**Tip:** When solving angle problems, breaking down complex angles into their components and using algebra to solve for unknown variables is often helpful!

Given m∠ABD = 96°, m∠CBD = 6x + 27°, and m∠ABC = 7x - 9°, find m∠CBD.

This geometry problem explores how to find the measure of angle m∠CBD using algebraic equations and the Angle Addition Postulate. It is designed for students in Grades 8-10 and covers key concepts such as angle relationships and solving linear equations.

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