Prove that the bisectors of two vertical angles are collinear

To prove that the bisectors of two vertical angles are collinear, we need to proceed step by step and use geometric reasoning. Let's define the situation clearly first:

### Step 1: Define Vertical Angles
- Let two intersecting lines \( l_1 \) and \( l_2 \) intersect at a point \( O \), creating two pairs of vertical (opposite) angles.
- Label the angles: \(\angle AOB\) and \(\angle COD\) as one pair of vertical angles, and \(\angle BOD\) and \(\angle AOC\) as the other pair.

Since \(\angle AOB = \angle COD\) and \(\angle AOC = \angle BOD\) (vertical angles are always congruent), we know that both pairs of vertical angles are equal.

### Step 2: Define the Angle Bisectors
- Let the bisector of \(\angle AOB\) be the line \( m_1 \), and let the bisector of \(\angle COD\) be the line \( m_2 \). These bisectors split the respective angles into two equal halves.
- Similarly, let the bisectors of \(\angle AOC\) and \(\angle BOD\) be lines \( m_3 \) and \( m_4 \), respectively.

### Step 3: Use Symmetry of Vertical Angles
Vertical angles are formed by two intersecting lines, and the angle bisectors of such angles must share certain symmetric properties. Since \(\angle AOB = \angle COD\), the bisectors of these angles must divide the entire plane into symmetric regions. Therefore, the bisectors of these angles must be reflections of each other across the point of intersection \( O \).

### Step 4: Collinearity of Bisectors
Because the angle bisectors \( m_1 \) and \( m_2 \) of the vertical angles \(\angle AOB\) and \(\angle COD\) divide the angles symmetrically and each bisector splits the plane into equal halves, the two bisectors must lie along the same straight line. Therefore, the bisectors are collinear and pass through the point \( O \).

### Conclusion
The bisectors of two vertical angles are collinear because they are symmetric reflections across the point of intersection \( O \), which is the common vertex of the vertical angles. Hence, the bisectors must lie along the same line.

### Do you want more details, or do you have any questions?

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Here are 5 related questions to deepen your understanding:

1. What is the definition of vertical angles in geometry?
2. How does symmetry play a role in proving geometric theorems?
3. Can the bisectors of adjacent angles (non-vertical) be collinear as well?
4. How do you construct an angle bisector geometrically?
5. How can you prove that two angles are congruent using vertical angles?

**Tip:** When working with symmetry in geometry, consider reflections and rotations, as they often help in proving relationships between angles and lines.

Prove that the bisectors of two vertical angles are collinear.

Learn how to prove that the bisectors of two vertical angles are collinear using geometric reasoning. This problem involves key concepts like the Vertical Angles Theorem, angle bisectors, and symmetry. Suitable for students in Grades 9-11.

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