Overview

Angle chasing is about exhausting known relationships before adding new construction lines.

Key Ideas

• Use triangle angle sums and parallel line angles first.
• Look for cyclic quadrilaterals and isosceles triangles.
• Keep angles in terms of a few variables and solve.

Core Skills

Control Variables

Use as few variables as possible. Express new angles in terms of existing ones to avoid an unsolvable system.

Spot Cyclicity

If opposite angles sum to $180^\circ$ or equal angles subtend the same chord, mark the cyclic quadrilateral.

Use Parallel Lines Early

Parallel lines create alternate interior angles and can collapse a chase.

Worked Example

In a triangle, two angles are $x$ and $x+30^\circ$. Find the third angle.

The third angle is $180^\circ - (2x + 30^\circ) = 150^\circ - 2x$.

More Examples

Example 1: Exterior Angle

If an exterior angle is $120^\circ$ and one remote interior angle is $50^\circ$, find the other remote interior angle.

It is $120^\circ-50^\circ=70^\circ$.

Example 2: Cyclic Opposite Angles

In a cyclic quadrilateral, one angle is $83^\circ$. Find the opposite angle.

$97^\circ$.

Example 3: Isosceles Triangle

If $AB=AC$ and $\angle B=35^\circ$, find $\angle A$.

$\angle A=180^\circ-2\cdot 35^\circ=110^\circ$.

Strategy Checklist

• Mark given angles and parallel lines first.
• Look for cyclic quadrilaterals or isosceles triangles.
• Keep variables minimal and solve linear equations.

Common Pitfalls

• Introducing too many variables.
• Forgetting to use cyclic angle sums when a circle is present.
• Missing alternate interior angles created by parallels.

Practice Problems