Vertically Opposite Angles

Definition: Vertically opposite angles are the angles formed by two intersecting lines. They are opposite to each other and share a common vertex.

Explanation:
When two lines intersect each other, they form four angles at the point of intersection. The angles that are opposite to each other and do not share a side are called vertically opposite angles.

Example:
Let's consider two lines, line AB and line CD, intersecting at point O. The angles formed at the point O are:
- ∠AOB (angle formed by line AB)
- ∠BOC (angle formed by line BC)
- ∠COD (angle formed by line CD)
- ∠DOA (angle formed by line DA)

Properties of Vertically Opposite Angles:
1. Equal Measure: Vertically opposite angles are equal. This means that ∠AOB is equal to ∠COD and ∠BOC is equal to ∠DOA. Mathematically, we can represent this as:
∠AOB = ∠COD
∠BOC = ∠DOA

2. Proof: We can prove that vertically opposite angles are equal using the property of alternate angles. When two lines intersect, they form two pairs of alternate angles. By applying the alternate angle property, we can show that the vertically opposite angles are equal.

Visual Representation:
To visually represent the concept of vertically opposite angles, we can draw two intersecting lines and label the angles formed at the point of intersection. We can clearly observe that the angles opposite to each other have the same measure.

Conclusion:
In conclusion, when two lines intersect each other, the vertically opposite angles formed are equal. This property holds true for any intersecting lines and can be proved using the alternate angle property. Understanding and applying this concept is important in various geometric problems and proofs.