https://file.aiquickdraw.com/mathbot/file/17301355602443t5gmkp7.jpg

Let's examine the angles given and solve the missing values using properties like vertical angles, corresponding angles, and supplementary angles.

### Given:
- \( m\angle 10 = 77^\circ \)
- \( m\angle 7 = 47^\circ \)
- \( m\angle 16 = 139^\circ \)

### Step-by-Step Solution:

1. **Using \( m\angle 10 = 77^\circ \):**
   - **Vertical angle** with \( m\angle 13 \), so \( m\angle 13 = 77^\circ \).
   - **Corresponding angle** with \( m\angle 2 \), so \( m\angle 2 = 77^\circ \).
   - **Alternate interior angle** with \( m\angle 15 \), so \( m\angle 15 = 77^\circ \).

2. **Using \( m\angle 7 = 47^\circ \):**
   - **Vertical angle** with \( m\angle 6 \), so \( m\angle 6 = 47^\circ \).
   - **Corresponding angle** with \( m\angle 3 \), so \( m\angle 3 = 47^\circ \).
   - **Alternate interior angle** with \( m\angle 8 \), so \( m\angle 8 = 47^\circ \).

3. **Using \( m\angle 16 = 139^\circ \):**
   - **Vertical angle** with \( m\angle 17 \), so \( m\angle 17 = 139^\circ \).
   - **Corresponding angle** with \( m\angle 12 \), so \( m\angle 12 = 139^\circ \).
   - **Alternate interior angle** with \( m\angle 14 \), so \( m\angle 14 = 139^\circ \).

4. **Remaining Angles:**
   - \( m\angle 1 \) is supplementary to \( m\angle 2 \) because they are on a straight line, so:
     \[
     m\angle 1 = 180^\circ - 77^\circ = 103^\circ
     \]
   - \( m\angle 4 \) is supplementary to \( m\angle 3 \), so:
     \[
     m\angle 4 = 180^\circ - 47^\circ = 133^\circ
     \]
   - \( m\angle 5 \) is supplementary to \( m\angle 6 \), so:
     \[
     m\angle 5 = 180^\circ - 47^\circ = 133^\circ
     \]
   - \( m\angle 9 \) is a vertical angle to \( m\angle 10 \), so:
     \[
     m\angle 9 = 77^\circ
     \]
   - \( m\angle 11 \) is supplementary to \( m\angle 12 \), so:
     \[
     m\angle 11 = 180^\circ - 139^\circ = 41^\circ
     \]
   - \( m\angle 18 \) is supplementary to \( m\angle 17 \), so:
     \[
     m\angle 18 = 180^\circ - 139^\circ = 41^\circ
     \]

### Completed Table:

| Angle       | Measure |
|-------------|---------|
| \( m\angle 1 \)  | \( 103^\circ \)  |
| \( m\angle 2 \)  | \( 77^\circ \)   |
| \( m\angle 3 \)  | \( 47^\circ \)   |
| \( m\angle 4 \)  | \( 133^\circ \)  |
| \( m\angle 5 \)  | \( 133^\circ \)  |
| \( m\angle 6 \)  | \( 47^\circ \)   |
| \( m\angle 8 \)  | \( 47^\circ \)   |
| \( m\angle 9 \)  | \( 77^\circ \)   |
| \( m\angle 11 \) | \( 41^\circ \)   |
| \( m\angle 12 \) | \( 139^\circ \)  |
| \( m\angle 13 \) | \( 77^\circ \)   |
| \( m\angle 14 \) | \( 139^\circ \)  |
| \( m\angle 15 \) | \( 77^\circ \)   |
| \( m\angle 17 \) | \( 139^\circ \)  |
| \( m\angle 18 \) | \( 41^\circ \)   |

Would you like further clarification or a breakdown of each angle relation? 

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Here are five related questions that could further enhance your understanding:

1. What are vertical angles, and how are they used to find angle measures?
2. How do corresponding angles help in solving for unknown angle measures in parallel lines?
3. Can alternate interior angles provide all needed angles in parallel line setups?
4. What is the supplementary angle theorem, and how is it applied in geometry problems?
5. How can identifying straight-line relationships simplify angle calculations?

**Tip:** Always look for vertical and supplementary angles first when working with intersecting lines and parallel lines; they often provide quick solutions.

Fill in the table below given the following diagram. If m∠10 = 77°, m∠7 = 47°, m∠16 = 139°

Solve the geometry problem by finding missing angle measurements in a complex parallel line setup. Given m∠10 = 77°, m∠7 = 47°, and m∠16 = 139°, use properties of vertical, supplementary, corresponding, and alternate interior angles. Suitable for students in Grades 8-10.

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