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Let's solve the two examples shown in the image.

### Example 2: **Recognizing Coterminal Angles**
Determine whether the following pairs of angles are coterminal.

- **a.** \( \alpha = 120^\circ, \beta = -180^\circ \)
- **b.** \( \alpha = 20^\circ, \beta = 740^\circ \)

#### Explanation:
Coterminal angles are angles that share the same terminal side. To check if two angles are coterminal, you can add or subtract multiples of \( 360^\circ \) until the angles fall within the range \( 0^\circ \) to \( 360^\circ \).

1. **a.** \( \alpha = 120^\circ, \beta = -180^\circ \)

   First, find the equivalent positive angle for \( \beta \):
   \[
   -180^\circ + 360^\circ = 180^\circ
   \]
   Since \( 120^\circ \neq 180^\circ \), these angles are **not coterminal**.

2. **b.** \( \alpha = 20^\circ, \beta = 740^\circ \)

   For \( \beta = 740^\circ \), subtract \( 360^\circ \) until it falls within \( 0^\circ \) to \( 360^\circ \):
   \[
   740^\circ - 360^\circ = 380^\circ
   \]
   \[
   380^\circ - 360^\circ = 20^\circ
   \]
   Since \( 20^\circ = 20^\circ \), these angles are **coterminal**.

---

### Example 3: **Finding Measures of Coterminal Angles**
Determine the angle of the smallest possible positive measure that is coterminal with each of the following angles.

- **a.** \( 830^\circ \)
- **b.** \( -520^\circ \)

#### Explanation:
To find the smallest positive coterminal angle, subtract or add \( 360^\circ \) until the angle falls within \( 0^\circ \) and \( 360^\circ \).

1. **a.** \( 830^\circ \)

   Subtract \( 360^\circ \) until the angle is within \( 0^\circ \) and \( 360^\circ \):
   \[
   830^\circ - 360^\circ = 470^\circ
   \]
   \[
   470^\circ - 360^\circ = 110^\circ
   \]
   So, the smallest positive coterminal angle is **110°**.

2. **b.** \( -520^\circ \)

   Add \( 360^\circ \) until the angle is positive:
   \[
   -520^\circ + 360^\circ = -160^\circ
   \]
   \[
   -160^\circ + 360^\circ = 200^\circ
   \]
   So, the smallest positive coterminal angle is **200°**.

---

Would you like more details or have any questions on these solutions? Here are 5 follow-up questions:

1. What are the general steps to find coterminal angles?
2. Why do we add or subtract multiples of \( 360^\circ \) when dealing with coterminal angles?
3. Can two negative angles be coterminal? How would you check?
4. How would you find the coterminal angle of a radian measure instead of degrees?
5. What happens when you have a fractional angle, like \( 450.5^\circ \)? How do you handle it?

**Tip:** Coterminal angles differ by full rotations of \( 360^\circ \). You can always reduce or increase angles by multiples of \( 360^\circ \) to find equivalent angles.

Determine whether the following pairs of angles are coterminal and find the smallest possible positive coterminal angles for given values.

Learn to determine whether pairs of angles are coterminal and how to find the smallest positive coterminal angle. Includes step-by-step solutions for angles like 120°, -180°, 830°, and -520°. Ideal for students in Grades 10-12.

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