Concept Notes 1: Intersecting Lines, Linear Pairs & Vertically Opposite Angles
When two straight lines cross in a plane, they interact at a single point called the point of intersection, generating pairs of related angles.
1. Fundamental Angle Relationships at an Intersection
- Intersecting Lines: Two lines that share exactly one common point.
- Adjacent Angles: Two angles that have a common vertex, a common arm, and non-overlapping interiors.
- Linear Pair: A pair of adjacent angles whose non-common arms form a straight line.
- The sum of two angles in a linear pair is always $180^\circ$ (supplementary):
$$\angle 1 + \angle 2 = 180^\circ$$
- Vertically Opposite Angles: The two non-adjacent angles formed by two intersecting lines that lie directly opposite each other across the vertex.
2. Theorem: Vertically Opposite Angles are Equal
Theorem: If two lines intersect, the vertically opposite angles are equal.
Formal Proof:
Let two lines $AB$ and $CD$ intersect at point $O$, forming angles $\angle 1, \angle 2, \angle 3, \angle 4$.
- Ray $OA$ stands on straight line $CD$. Therefore, $\angle 1$ and $\angle 2$ form a linear pair:
$$\angle 1 + \angle 2 = 180^\circ \quad \text{--- (Equation 1)}$$
- Ray $OD$ stands on straight line $AB$. Therefore, $\angle 2$ and $\angle 3$ form a linear pair:
$$\angle 2 + \angle 3 = 180^\circ \quad \text{--- (Equation 2)}$$
- Equating both sums:
$$\angle 1 + \angle 2 = \angle 2 + \angle 3$$
- Subtracting $\angle 2$ from both sides:
$$\mathbf{\angle 1 = \angle 3}$$
By identical logic on rays $OB$ and $OC$, we prove that $\mathbf{\angle 2 = \angle 4}$.