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.