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Imagine opening a pair of scissors: as the blades widen, the handles widen by the exact same amount. When two straight lines cross, the geometry locks every angle into a strict relationship.
If we replace those angle values with algebra, we can decode every unknown measurement with simple linear equations.

The Two Intersecting Rules

When two straight lines intersect, they form four angles meeting at a single point called the vertex. These angles always form two core relationships.
First, vertical angles sit opposite each other across the vertex and are always equal in measure. Over 2,600 years ago, the Greek mathematician Thales of Miletus was the first to mathematically prove this relationship.
Second, adjacent angles that sit side-by-side along a straight line form a linear pair and are supplementary, meaning their measures add up to 180∘180^\circ.
What happens when the opposite angles are written as algebraic expressions instead of fixed numbers?

Solving Vertical Angle Equations

Since vertical angles are always congruent (equal in size), you set their algebraic expressions equal to each other: Expression 1=Expression 2\text{Expression } 1 = \text{Expression } 2.
Let's solve an example: two vertical angles are labeled (5x−10)∘(5x - 10)^\circ and (3x+20)∘(3x + 20)^\circ.
Always remember that solving for x=15x = 15 is only the first step — plugging 1515 back into either expression reveals that both angles measure 65∘65^\circ.