Let's analyze the mathematical problem in the image step by step.

Problem Analysis:

The question involves two diagrams and asks about angular relationships and calculations:

1. Given Information:

   • $\angle AOC = 2 \angle BOC$
   • Points $A, O, B$ are collinear.
   • Various angles and rotations are given, and the relationship between them needs to be analyzed.

2. Tasks:

   1. Solve for $\angle EOB$ when line $OD$ bisects $\angle AOC$ and $\angle DOE = 80^\circ$.
   2. Analyze the relationship between $\angle FOE$ and $\angle EOC$ after rotation involving $\angle FOA$ and $\angle AOD$.
   3. Solve for the values of angles when specific rotational and angular relationships are defined.

Let's Solve:

1. For $\angle EOB$ when $OD$ bisects $\angle AOC$:

• Given: $\angle AOC = 2 \angle BOC$.
• Let $\angle BOC = x$. Then, $\angle AOC = 2x$.
• Since $OD$ bisects $\angle AOC$: $\angle AOD = \angle DOC = x.$
• $\angle DOE = 80^\circ$. This angle lies in the region above line $AB$.
• $\angle EOB$ can be calculated using: $\angle EOB = 180^\circ - (\angle DOE + \angle BOC).$
• Substituting $\angle DOE = 80^\circ$ and $\angle BOC = x$: $\angle EOB = 180^\circ - (80^\circ + x) = 100^\circ - x.$

2. For $\angle FOE$ and $\angle EOC$:

• Given:
  • $\angle FOA = 3 \angle AOD$.
  • Rotation of line $OF$ through $n^\circ (0 < n < 60)$.
• Substitute the relationships: $\angle FOA = 3x \quad \text{(since } \angle AOD = x\text{)}.$
• Analyze $\angle FOE$ based on rotation $n^\circ$. The relationship depends on whether the rotation adds or subtracts from $\angle FOA$.

3. For the second diagram ($OB$ edge rotation):

• Start with $\angle AOB < 108^\circ$.
• Given relationships:
  • $\angle DOE = \frac{1}{2} \angle AOB$,
  • When OB rotates, we are tasked with finding $\angle AOD + \angle EOC = \frac{3}{2} \angle BOE$.
• Detailed angular relationships will require solving for $BOE$ first.

Would you like to go into detailed calculations for each part or focus on a specific one?

Related Questions:

1. How does bisecting an angle relate to linear or rotational geometry?
2. What are alternative methods to calculate $\angle EOB$ if additional constraints were provided?
3. How do rotational transformations affect angle relationships in diagrams?
4. Could you solve $\angle FOE$ geometrically instead of algebraically?
5. What are practical applications of these angular relationships in fields like engineering or physics?

Tip:

Always label angles with variables and write relationships explicitly; this helps avoid errors during algebraic manipulation.