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The problem asks to prove that the bisectors of two vertical angles are collinear, meaning they lie on the same line.

### Proof Outline:
1. **Definition of Vertical Angles:**
   Vertical angles are formed when two straight lines intersect. Let’s assume that two lines intersect at point \( O \), forming four angles. The pairs of vertical angles are congruent (equal in measure). Let these angles be \( \angle AOB \) and \( \angle COD \).

2. **Angle Bisectors:**
   The angle bisectors are the rays that divide each of these angles into two equal parts. Let \( m \) be the bisector of \( \angle AOB \) and \( n \) be the bisector of \( \angle COD \).

3. **Equal Angles on Both Sides:**
   Since \( \angle AOB = \angle COD \) (as they are vertical angles), the bisectors of these angles divide them into two equal halves. Therefore, both the bisectors of \( \angle AOB \) and \( \angle COD \) form adjacent angles that sum up to 180°.

4. **Collinearity of the Bisectors:**
   The bisectors \( m \) and \( n \) divide the plane into congruent adjacent angles. Since these two adjacent angles formed by the bisectors sum to 180°, the two bisectors must lie on the same line. Therefore, the bisectors of two vertical angles are collinear.

Thus, the bisectors of two vertical angles are collinear because they divide the intersecting angles into equal halves, and these halves are supplementary, summing up to 180°.

Would you like further details or clarifications on any part of this proof?

Here are some related questions to explore:
1. What are vertical angles and how are they formed?
2. Can the bisectors of non-vertical angles also be collinear?
3. How does the congruency of vertical angles influence the bisectors?
4. What is the geometric significance of angle bisectors?
5. How do we generalize this proof for more complex intersection angles?

**Tip:** When working with angle bisectors, remember that the sum of adjacent angles formed by the bisectors can help determine whether they are collinear.

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

This page offers a detailed proof that the bisectors of two vertical angles are collinear. Learn about vertical angles, the Vertical Angles Theorem, and how the bisectors of congruent angles lie on the same line. Ideal for students in Grades 9-11.

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