https://file.aiquickdraw.com/mathbot/file/1726502607839foazq6zq.jpg

Let's solve the problem step by step. The diagram shows a triangle \( ACD \) and a larger triangle \( ABD \), and we are asked to find the angle \( CDB \).

### Step 1: Identify Known Information
- The triangle \( ACD \) is a smaller triangle within \( ABD \).
- The angle \( \angle ACD = 40^\circ \).
- The angle at \( A \) is \( 2x \).
- The angle \( \angle DCB = x + 12 \).
- The angle at \( B \), \( \angle ABD = x \).

### Step 2: Use Triangle Angle Sum in \( \triangle ACD \)
The sum of angles in a triangle is \( 180^\circ \). For triangle \( ACD \):
\[
\angle ACD + \angle CAD + \angle CDA = 180^\circ
\]
Substituting the values:
\[
40^\circ + 2x + (x + 12) = 180^\circ
\]
Simplify this equation:
\[
52^\circ + 3x = 180^\circ
\]
Solve for \( x \):
\[
3x = 180^\circ - 52^\circ
\]
\[
3x = 128^\circ
\]
\[
x = \frac{128^\circ}{3} = 42.67^\circ
\]

### Step 3: Calculate \( \angle CDB \)
Now, use \( x \) to find \( \angle CDB \). Since \( \angle CDB = x \), we substitute \( x = 42.67^\circ \) into the equation:
\[
\angle CDB = 42.67^\circ
\]

Thus, the size of angle \( CDB \) is approximately **42.67 degrees**.

Do you have any questions or need further clarification? 

Here are five related questions to explore:
1. What is the sum of the internal angles in any triangle?
2. How do you determine the value of a variable in a geometric equation?
3. What is the relationship between angles in a straight line?
4. Can you explain how supplementary angles work?
5. How can angles be solved when dealing with multiple variables?

**Tip:** Always verify the sum of angles in a triangle adds up to 180 degrees when solving problems involving triangles.

Work out the size of angle CDB, given the angles in triangle ACD and ABD, including algebraic expressions for some of the angles.

Learn how to calculate the size of angle CDB in a triangle using algebraic expressions for the angles in triangles ACD and ABD. This problem demonstrates the Triangle Angle Sum Theorem and solving for unknown variables. Suitable for students in Grade 8-10.

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