The given image shows a triangle $\triangle ABC$ with angles represented algebraically as $80^\circ, 2x + 20$, and $3x + 10$. The problem asks to find the value of $x$, and the measures of $\angle CBF$ and $\angle DCB$.

Let's break down the solution step-by-step:

(a) Finding $x$:

The sum of angles in any triangle is $180^\circ$.

$80 + (2x + 20) + (3x + 10) = 180$

Simplifying:

$80 + 2x + 20 + 3x + 10 = 180$

$5x + 110 = 180$

$5x = 180 - 110$

$5x = 70$

$x = 14$

(b) $\angle CBF$:

We need to calculate $\angle CBF$. The information for this angle isn't clear from the given image, but let's assume the problem asks for an exterior angle or adjacent angle. Since $\angle CBF$ is not directly stated, more context is needed to clarify this.

(c) $\angle DCB$:

Similarly, to find $\angle DCB$, the diagram doesn't give enough details. Let's calculate $\angle DBC$.

For now, let's assume the given angles and sides:

• For $\angle BAC = 2x + 20 = 2(14) + 20 = 28 + 20 = 48^\circ$.
• For $\angle ABC = 3x + 10 = 3(14) + 10 = 42 + 10 = 52^\circ$.
• $\angle ACB = 180 - (80 + 48) = 180 - 128 = 52^\circ$.

If you have more details about $\angle CBF$ and $\angle DCB$, please provide them.

Do you have any specific questions, or would you like more details on a particular part?

Relative Questions:

1. What is the significance of $\angle CBF$ and $\angle DCB$ in the context of this triangle?
2. Are there any other properties of $\triangle ABC$ that we should consider (e.g., is it isosceles)?
3. How would we verify the triangle angle sum property using this data?
4. Can we find side lengths or relationships using the angle measures given?
5. Are there any exterior angles we need to account for?

Tip: Always recheck your calculated angle measures to ensure they align with triangle properties, such as the sum being 180 degrees.