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Let's solve these three problems step by step by finding the exterior angles.

### **Problem 6:**
- Interior angles: \( (2x - 25)^\circ \) and \( x^\circ \)
- Exterior angle: \( 20^\circ \)

Using the **exterior angle theorem**:
\[
\text{Exterior Angle} = \text{Sum of Remote Interior Angles}
\]
\[
20^\circ = (2x - 25)^\circ + x^\circ
\]
\[
20^\circ = 3x - 25^\circ
\]
Add \(25^\circ\) to both sides:
\[
45^\circ = 3x
\]
Divide by 3:
\[
x = 15^\circ
\]

So, \( x = 15^\circ \).

---

### **Problem 7:**
- Interior angles: \( 2x^\circ \) and \( 8x^\circ \)  
- Note: The sum of interior angles of a triangle is \(180^\circ\).

We need to find the exterior angle adjacent to the right angle, which is also an exterior angle at the bottom vertex.  
\[
2x^\circ + 8x^\circ + 90^\circ = 180^\circ
\]
\[
10x + 90 = 180
\]
Subtract 90 from both sides:
\[
10x = 90
\]
Divide by 10:
\[
x = 9^\circ
\]

So, \(x = 9^\circ\).  
Thus, the exterior angle is the supplement of \(8x = 8(9) = 72^\circ\):
\[
180^\circ - 72^\circ = 108^\circ
\]

---

### **Problem 8:**
- Interior angles: \(45^\circ\) and \(5x^\circ\)
- Exterior angle: \(9x + 9^\circ\)

Using the **exterior angle theorem**:
\[
9x + 9^\circ = 45^\circ + 5x^\circ
\]
\[
9x + 9^\circ = 45^\circ + 5x^\circ
\]
Subtract \(5x\) from both sides:
\[
4x + 9^\circ = 45^\circ
\]
Subtract 9 from both sides:
\[
4x = 36^\circ
\]
Divide by 4:
\[
x = 9^\circ
\]

---

### **Summary:**
- **Problem 6**: \(x = 15^\circ\)
- **Problem 7**: Exterior angle = \(108^\circ\)
- **Problem 8**: \(x = 9^\circ\)

Would you like any part of the solution explained in more detail?

---

Here are 5 follow-up questions to practice:

1. How does the exterior angle theorem apply to all types of triangles?
2. How do you find the third angle of a triangle if you know two angles?
3. What is the relationship between interior and exterior angles in a polygon?
4. Can an exterior angle ever be smaller than any interior angle? Why or why not?
5. How do supplementary angles relate to exterior angles?

**Tip:** The sum of the interior angles of any triangle is always \(180^\circ\).

Find the measure of the exterior angle shown in problems 6, 7, and 8.

Learn how to find the measure of exterior angles in triangles using the Exterior Angle Theorem. This exercise covers problems involving algebraic expressions for angle measures and is suitable for students in Grades 8-10.

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