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To prove that the bisectors of two vertical angles are collinear (lie on the same line), we need to follow a step-by-step geometric approach. Let’s go through the proof.

### Given:
- Two vertical angles, \( \angle AOB \) and \( \angle COD \), which are formed when two straight lines intersect at point \( O \).
- Bisectors of these angles are drawn.

### To Prove:
- The bisectors of the vertical angles are collinear, meaning they lie on the same line.

### Proof:
1. **Definition of Vertical Angles**:  
   Vertical angles are formed when two lines intersect. Let line \( AB \) and line \( CD \) intersect at point \( O \). This forms two pairs of vertical angles:
   - \( \angle AOB \) and \( \angle COD \)
   - \( \angle AOC \) and \( \angle BOD \)

2. **Properties of Vertical Angles**:  
   From the property of vertical angles, we know that:
   \[
   \angle AOB = \angle COD \quad \text{and} \quad \angle AOC = \angle BOD
   \]
   This is because vertical angles are congruent.

3. **Construct the Bisectors**:  
   Let \( OE \) be the bisector of \( \angle AOB \), and let \( OF \) be the bisector of \( \angle COD \). By the definition of an angle bisector, the bisector divides an angle into two equal parts:
   \[
   \angle AOE = \angle BOE \quad \text{and} \quad \angle COF = \angle DOF
   \]

4. **Measure of Half-Angles**:  
   Since \( \angle AOB = \angle COD \), and the bisectors divide these angles into equal parts, we have:
   \[
   \angle AOE = \angle COF \quad \text{and} \quad \angle BOE = \angle DOF
   \]
   Therefore, both \( OE \) and \( OF \) divide the vertical angles symmetrically.

5. **Collinearity of Bisectors**:  
   Since \( OE \) is the bisector of \( \angle AOB \) and \( OF \) is the bisector of \( \angle COD \), and both \( \angle AOB \) and \( \angle COD \) are congruent vertical angles, the bisectors \( OE \) and \( OF \) must lie along the same line.

   Therefore, the bisectors of the two vertical angles are collinear.

### Conclusion:
The bisectors of two vertical angles are collinear because they symmetrically divide congruent angles and lie along the same line.

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Would you like further clarification or more details on any specific step?

Here are 5 related questions for further exploration:
1. What is the general property of vertical angles in terms of their measurements?
2. How do you prove that two angles are congruent in a general case?
3. Can you extend this proof to other types of angles, such as adjacent angles?
4. What happens if the intersecting lines are perpendicular—what is the angle between their bisectors?
5. How can this concept of collinearity of bisectors be applied to other geometric shapes?

**Tip:** Always remember that vertical angles are congruent, and their bisectors divide them symmetrically, leading to interesting results like collinearity.

Prove that the bisectors of two vertical angles are collinear (lie on the same line).

This proof demonstrates that the bisectors of two vertical angles, which are congruent, lie on the same line. It explores properties of vertical angles and angle bisectors using fundamental geometric theorems, suitable for students in Grades 10-12.

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