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.

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 $180^\circ$.