https://file.aiquickdraw.com/mathbot/file/1729829549143kf0hfehl.jpg

Let's solve this problem step by step.

We are given:
- \( m\angle 9 = 97^\circ \)
- \( m\angle 12 = 114^\circ \)

### Observations:
- The angles are labeled around intersecting lines. This means we will use properties of vertical angles, corresponding angles, and supplementary angles.

### Using the angle properties:

1. **Vertical Angles**: Vertical angles are equal.
   - \( m\angle 9 \) and \( m\angle 11 \) are vertical angles, so \( m\angle 11 = 97^\circ \).
   - \( m\angle 12 \) and \( m\angle 14 \) are vertical angles, so \( m\angle 14 = 114^\circ \).

2. **Supplementary Angles**: Angles on a straight line add up to \( 180^\circ \).
   - \( m\angle 9 + m\angle 10 = 180^\circ \), so \( m\angle 10 = 180^\circ - 97^\circ = 83^\circ \).
   - \( m\angle 12 + m\angle 16 = 180^\circ \), so \( m\angle 16 = 180^\circ - 114^\circ = 66^\circ \).

3. **Corresponding Angles**: Corresponding angles are equal when lines are parallel.
   - \( m\angle 9 = m\angle 1 \), so \( m\angle 1 = 97^\circ \).
   - \( m\angle 12 = m\angle 4 \), so \( m\angle 4 = 114^\circ \).

4. **Alternate Interior Angles**: Alternate interior angles are equal when lines are parallel.
   - \( m\angle 10 = m\angle 15 \), so \( m\angle 15 = 83^\circ \).
   - \( m\angle 11 = m\angle 16 \), which we already calculated as \( 66^\circ \).

### Now, filling in the table:

- a. \( m\angle 1 = 97^\circ \) (corresponding angle to \( m\angle 9 \))
- b. \( m\angle 2 = 83^\circ \) (supplementary to \( m\angle 1 \), forming a straight line)
- c. \( m\angle 3 = 97^\circ \) (vertical angle to \( m\angle 2 \))
- d. \( m\angle 4 = 114^\circ \) (corresponding angle to \( m\angle 12 \))
- e. \( m\angle 5 = 66^\circ \) (supplementary to \( m\angle 4 \))
- f. \( m\angle 6 = 114^\circ \) (vertical angle to \( m\angle 5 \))
- g. \( m\angle 7 = 66^\circ \) (alternate interior angle to \( m\angle 6 \))
- h. \( m\angle 8 = 114^\circ \) (supplementary to \( m\angle 7 \))
- i. \( m\angle 10 = 83^\circ \) (from earlier calculation)
- j. \( m\angle 11 = 97^\circ \) (vertical angle to \( m\angle 9 \))
- k. \( m\angle 13 = 66^\circ \) (vertical angle to \( m\angle 14 \))
- l. \( m\angle 14 = 114^\circ \) (given)
- m. \( m\angle 15 = 83^\circ \) (alternate interior angle to \( m\angle 10 \))
- n. \( m\angle 16 = 66^\circ \) (from earlier calculation)

Would you like further clarification on any step? 

### Related Questions:
1. What are the properties of vertical angles and how can they be used in geometric problems?
2. How do supplementary angles form on a straight line, and how is this applied in parallel lines?
3. What is the significance of alternate interior angles in parallel lines?
4. Can you explain how corresponding angles are used when working with parallel lines?
5. How does knowing one angle in intersecting lines help find all other angles?

**Tip**: Always check if angles are vertically opposite or supplementary when dealing with intersecting lines—this greatly simplifies the solution!

If m∠9 = 97° and m∠12 = 114°, find each measure (based on the image provided).

This geometry problem focuses on finding unknown angle measures given m∠9 = 97° and m∠12 = 114°. The solution involves using properties of vertical angles, supplementary angles, corresponding angles, and alternate interior angles, ideal for students in Grades 8-10.

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