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The problem presented asks to prove that vertically opposite angles are equal when two lines intersect. In the image, lines \( AB \) and \( CD \) intersect at point \( O \), forming four angles \( \angle 1 \), \( \angle 2 \), \( \angle 3 \), and \( \angle 4 \).

### Proof:

1. When two lines intersect at point \( O \), four angles are formed: \( \angle 1 \), \( \angle 2 \), \( \angle 3 \), and \( \angle 4 \).

2. From geometry, we know that adjacent angles formed by intersecting lines are **supplementary**, meaning their sum equals \( 180^\circ \). Therefore:
   \[
   \angle 1 + \angle 2 = 180^\circ \quad \text{(linear pair)}
   \]
   \[
   \angle 3 + \angle 4 = 180^\circ \quad \text{(linear pair)}
   \]

3. Also, notice that the sum of angles on the opposite side of each line is also \( 180^\circ \). Therefore:
   \[
   \angle 1 + \angle 4 = 180^\circ
   \]
   \[
   \angle 2 + \angle 3 = 180^\circ
   \]

4. By equating the sums of angles from the same line, we find that \( \angle 1 = \angle 3 \) and \( \angle 2 = \angle 4 \). This is because the only way for all angles around point \( O \) to sum up to \( 360^\circ \) while maintaining linear pairs is if the opposite angles are equal.

Thus, **vertically opposite angles** \( \angle 1 = \angle 3 \) and \( \angle 2 = \angle 4 \), proving the statement.

Would you like more details or have any questions?

### Related Questions:
1. What is the definition of supplementary angles?
2. How does the property of linear pairs relate to the angle sum around a point?
3. Can this proof be extended to non-Euclidean geometries?
4. What other properties arise from the intersection of two lines?
5. How would you prove the equality of vertical angles using trigonometry?

### Tip:
Always check if the angles form linear pairs when dealing with intersecting lines, as this is key to proving vertical angles are equal.

If two lines intersect each other, then the vertically opposite angles are equal. Prove that.

Learn how to prove that vertically opposite angles are equal when two lines intersect. This geometric proof involves understanding linear pairs and the properties of angles formed by intersecting lines. Suitable for students in Grades 6-8.

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