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.