Let's solve the given problem step by step.

Given Data:

1. The outer angles of the triangle are:
   • $80^\circ$
   • $2x + 20^\circ$
   • $3x + 10^\circ$

Part (a): Calculate the value of $x$.

Using the fact that the sum of the outer angles of any triangle is always $360^\circ$, we can set up the equation:

$80 + (2x + 20) + (3x + 10) = 360$

Now, let's solve this equation for $x$.

$80 + 2x + 20 + 3x + 10 = 360$ $110 + 5x = 360$ $5x = 360 - 110$ $5x = 250$ $x = \frac{250}{5} = 50$

So, $x = 50$.

Part (b): What is the measure of $\angle CBF$?

From the diagram, $\angle CBF$ is given as $2x + 20$. Now that we know $x = 50$, we can substitute this value into the expression for $\angle CBF$:

$\angle CBF = 2(50) + 20 = 100 + 20 = 120^\circ$

Thus, $\angle CBF = 120^\circ$.

Part (c): What is the measure of $\angle DCB$?

Similarly, $\angle DCB$ is given as $3x + 10$. Substituting $x = 50$ into this expression:

$\angle DCB = 3(50) + 10 = 150 + 10 = 160^\circ$

Thus, $\angle DCB = 160^\circ$.

Summary of Results:

• (a) $x = 50$
• (b) $\angle CBF = 120^\circ$
• (c) $\angle DCB = 160^\circ$

Do you need further details or have any questions about this solution?

Here are 5 related questions to practice:

1. What is the sum of the interior angles of a triangle?
2. If one outer angle of a triangle is $x + 30$, another is $2x + 40$, and the third is 80°, find $x$.
3. What is the relationship between an interior and exterior angle of a triangle?
4. Can a triangle have more than one obtuse outer angle? Why or why not?
5. How does the sum of the outer angles of a polygon change as the number of sides increases?

Tip: Remember, the sum of the exterior angles of any polygon, regardless of the number of sides, is always 360°.