Imagine opening a pair of scissors. As one pair of scissor blades opens wider, what happens to the opposite blades? They open by the exact same angle! This is the geometric guarantee of vertically opposite angles.
- Linear Pair Axiom: If a ray stands on a line, then the sum of two adjacent angles so formed is $180^\circ$.
- Theorem 6.1 (Vertically Opposite Angles): If two lines intersect each other, then the vertically opposite angles are equal ($\angle 1 = \angle 3$ and $\angle 2 = \angle 4$).
Problem: Lines $AB$ and $CD$ intersect at $O$. If $\angle AOC + \angle BOE = 70^\circ$ and $\angle BOD = 40^\circ$, find $\angle BOE$ and reflex $\angle COE$.
Step 1: $\angle AOC = \angle BOD = 40^\circ$ (Vertically Opposite Angles).
Step 2: $\angle AOC + \angle BOE = 70^\circ \implies 40^\circ + \angle BOE = 70^\circ \implies \angle BOE = 30^\circ$.
Step 3: $AOB$ is a line $\implies \angle AOC + \angle COE + \angle BOE = 180^\circ \implies 40^\circ + \angle COE + 30^\circ = 180^\circ \implies \angle COE = 110^\circ$.
Reflex $\angle COE$: $360^\circ - 110^\circ = 250^\circ$.
When asked for "reflex $\angle COE$", students often simply report the acute/obtuse interior angle ($110^\circ$)!
Rule: The reflex of any angle $\theta$ is strictly $360^\circ - \theta$. Reflex $\angle COE = 360^\circ - 110^\circ = 250^\circ$.
Snooker and billiards players calculate cushion bank shots using the optical reflection law $\angle i = \angle r$, a direct application of supplementary normal angles.