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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°.

In the figure, the outer angles of a triangle ABC are 80°, 2x + 20°, and 3x + 10°. (a) Calculate the value of x. (b) What is the measure of ∠CBF? (c) What is the measure of ∠DCB?

This math problem involves calculating the value of x and determining the angles in a triangle using the sum of exterior angles theorem. Learn how to apply this concept with step-by-step solutions. Suitable for students in Grades 7-10.

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