Let's work through the problems in the image step-by-step.

1. Solving for the missing angle values:

(a) Angles:

• m∠1 = ?
• m∠2 = ?
• m∠3 = ?
• m∠4 = ?

The given angles are:

• m∠1 + 33° = 180° (linear pair)
  • m∠1 = 180° - 33° = 147°
• m∠2 + m∠3 = 180° (opposite angles)
  • Since m∠3 = 59°, m∠2 = 180° - 59° = 121°
• m∠4 = m∠1 (opposite angles)
  • m∠4 = 147°

So the missing angles are:

• m∠1 = 147°
• m∠2 = 121°
• m∠3 = 59°
• m∠4 = 147°

(b) Angles:

• m∠1 = ?
• m∠2 = ?
• m∠3 = ?
• m∠4 = ?

Given:

• m∠3 = 47°
  • Opposite angles are equal, so m∠1 = 47°
• m∠2 + m∠3 = 180° (linear pair)
  • m∠2 = 180° - 47° = 133°
• m∠4 = m∠2 (opposite angles)
  • m∠4 = 133°

So the missing angles are:

• m∠1 = 47°
• m∠2 = 133°
• m∠3 = 47°
• m∠4 = 133°

(c) Angles:

• m∠1 = ?
• m∠2 = ?
• m∠3 = ?

Given:

• m∠1 = 37°
  • m∠2 and m∠3 are supplementary to 90° because they are adjacent to a right angle.
  • m∠2 = 90° - m∠1 = 90° - 37° = 53°
  • m∠3 = 90° (because it's a right angle)

So the missing angles are:

• m∠1 = 37°
• m∠2 = 53°
• m∠3 = 90°

2. Solving for x:

First diagram:

• $6x^\circ + 53^\circ = 180^\circ$ (since these angles form a linear pair)
  • $6x = 180^\circ - 53^\circ = 127^\circ$
  • $x = \frac{127^\circ}{6} \approx 21.17^\circ$

Second diagram:

• $4x + 54^\circ = 180^\circ$ (again, these form a linear pair)
  • $4x = 180^\circ - 54^\circ = 126^\circ$
  • $x = \frac{126^\circ}{4} = 31.5^\circ$

Third diagram:

• $8x + 2x + 17^\circ = 90^\circ$ (because they form a right angle)
  • $10x + 17^\circ = 90^\circ$
  • $10x = 90^\circ - 17^\circ = 73^\circ$
  • $x = \frac{73^\circ}{10} = 7.3^\circ$

Would you like further details or explanations for any of these steps?

Here are five related questions:

1. How do we find missing angles in other types of intersecting lines?
2. What is the significance of supplementary and complementary angles?
3. Can you explain how vertical angles work in more complex geometries?
4. How can we generalize the use of linear pairs in solving for unknowns?
5. How can systems of equations help solve for unknown angle measures?

Tip: Always check for angle relationships like linear pairs, vertical angles, and supplementary angles when solving problems involving intersecting lines!