https://file.aiquickdraw.com/mathbot/file/1726588483906yjstsmzu.jpg

Let's solve the problems in the image you've uploaded:

### 1. Given:
- \( m\angle ABD = 40^\circ \)
- \( m\angle ABC = 78^\circ \)

Find \( m\angle DBC \):

- We know that \( m\angle ABC = m\angle ABD + m\angle DBC \).
- So, \( 78^\circ = 40^\circ + m\angle DBC \).
- Solving for \( m\angle DBC \), we get \( m\angle DBC = 78^\circ - 40^\circ = 38^\circ \).

### 2. Given:
- \( m\angle ABD = 40^\circ \)
- \( m\angle DBC = 36^\circ \)

Find \( m\angle ABC \):

- We know that \( m\angle ABC = m\angle ABD + m\angle DBC \).
- So, \( m\angle ABC = 40^\circ + 36^\circ = 76^\circ \).

### 3. Given:
- \( m\angle WXZ = 57^\circ \)
- \( m\angle WXY = 123^\circ \)

Find \( m\angle ZXY \):

- We know that \( m\angle WXY = m\angle WXZ + m\angle ZXY \).
- So, \( 123^\circ = 57^\circ + m\angle ZXY \).
- Solving for \( m\angle ZXY \), we get \( m\angle ZXY = 123^\circ - 57^\circ = 66^\circ \).

### 4. Given:
- \( m\angle ZXY = 54^\circ \)
- \( m\angle WXZ = 60^\circ \)

Find \( m\angle WXY \):

- We know that \( m\angle WXY = m\angle WXZ + m\angle ZXY \).
- So, \( m\angle WXY = 60^\circ + 54^\circ = 114^\circ \).

### 5. Given:
- \( m\angle LMN = 135^\circ \)
- \( m\angle LMV = 45x - 1^\circ \)
- \( m\angle VMN = 23x \)

Find \( x \):

- The three angles form a straight line, so their sum should be \( 180^\circ \).
- Set up the equation: 
  \[
  135^\circ + (45x - 1^\circ) + 23x = 180^\circ.
  \]
- Simplifying:
  \[
  135 + 45x - 1 + 23x = 180,
  \]
  \[
  134 + 68x = 180,
  \]
  \[
  68x = 46,
  \]
  \[
  x = \frac{46}{68} = \frac{23}{34} \approx 0.676.
  \]

### 6. Given:
- \( m\angle GFE = 157^\circ \)
- \( m\angle NFE = 14x + 3^\circ \)
- \( m\angle GFN = 4x + 10^\circ \)

Find \( x \):

- The sum of the three angles should be \( 180^\circ \) because they form a straight line.
- Set up the equation:
  \[
  157^\circ + (14x + 3^\circ) + (4x + 10^\circ) = 180^\circ,
  \]
  \[
  157 + 14x + 3 + 4x + 10 = 180,
  \]
  \[
  170 + 18x = 180,
  \]
  \[
  18x = 10,
  \]
  \[
  x = \frac{10}{18} = \frac{5}{9} \approx 0.556.
  \]

### 7. Classify the angles as acute, obtuse, right, or straight:
- \( \angle DOG \): Appears **obtuse** (greater than \(90^\circ\)).
- \( \angle DOS \): Appears **acute** (less than \(90^\circ\)).
- \( \angle DSG \): Appears **obtuse**.
- \( \angle GOS \): Appears **acute**.

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Would you like more details on any specific step or have any questions? 

Here are 5 related questions to think about:
1. How would you classify angles based on their measurements?
2. How can you find an unknown angle in a triangle given two other angles?
3. What is the relationship between adjacent angles on a straight line?
4. How do you solve for \(x\) in an equation with angles forming a straight line?
5. What is the sum of interior angles of a polygon?

**Tip**: Always remember that the sum of angles on a straight line is \(180^\circ\).

Find unknown angles in geometry problems and classify angles as acute, obtuse, right, or straight.

Learn how to solve problems involving supplementary angles, including angle addition and finding unknown angles. This step-by-step solution is perfect for middle school students (Grades 6-8) exploring geometry and algebra concepts.

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