https://file.aiquickdraw.com/mathbot/file/17266870570636zsllhmw.jpg

Let's analyze the given geometric figure and solve the problem step by step.

### Analysis:
- The angle \( \angle ABC \) is composed of two parts: \( (5x + 11)^\circ \) and \( (4x + 16)^\circ \), which are adjacent angles.
- \( \angle ABC \) forms a right angle with \( \overrightarrow{BA} \), meaning \( \angle ABC = 90^\circ \).
- The sum of the angles \( \angle ABC = (5x + 11)^\circ + (4x + 16)^\circ \).

### Step 1: Solve for \( x \)
We know that the total measure of \( \angle ABC \) must be \( 90^\circ \). Therefore, we can set up the equation:
\[
(5x + 11) + (4x + 16) = 90
\]

Simplify the equation:
\[
9x + 27 = 90
\]

Subtract 27 from both sides:
\[
9x = 63
\]

Now divide by 9:
\[
x = 7
\]

### Step 2: Calculate \( m\angle ABC \)
We already know that \( m\angle ABC = 90^\circ \) because it forms a right angle.

### Step 3: Calculate \( m\angle CBD \)
To find \( m\angle CBD \), substitute \( x = 7 \) into the expression for the angle \( (4x + 16) \):
\[
m\angle CBD = 4(7) + 16 = 28 + 16 = 44^\circ
\]

### Final Answers:
- \( x = 7 \)
- \( m\angle ABC = 90^\circ \)
- \( m\angle CBD = 44^\circ \)

Would you like further details or clarifications on any step? 

Here are some related questions for further practice:
1. What would happen if \( \angle ABC \) was an obtuse angle instead of a right angle?
2. How would the solution change if \( \angle ABC \) were split into three parts?
3. Can you find the value of \( m\angle ABC \) if the expressions for the angles were different?
4. What is the significance of the sum of adjacent angles in geometry?
5. How would you solve this if the diagram represented supplementary angles?

**Tip:** Always check if the angle relationships, like adjacent or complementary, help simplify the equation setup!

Given that angle ABC consists of two adjacent angles (5x + 11)° and (4x + 16)°, find x, the measure of angle ABC, and angle CBD.

Learn how to solve for x and calculate the measures of angles ABC and CBD in a right triangle using algebraic and geometric concepts. This step-by-step solution explains how to apply the adjacent angles theorem and solve linear equations. Suitable for Grades 7-10.

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