Angle chasing isn't really one technique — it's knowing this short list of facts well enough to spot which one applies at each step of a diagram:
- Angles in a triangle sum to .
- Exterior angle theorem: an exterior angle of a triangle equals the sum of the two non-adjacent interior angles.
- Parallel lines cut by a transversal: corresponding angles are equal, alternate interior angles are equal, and co-interior (same-side interior) angles are supplementary.
- Inscribed angle theorem: an inscribed angle equals half its intercepted arc.
- An angle inscribed in a semicircle is always .
- Opposite angles of a cyclic quadrilateral are supplementary.
The inscribed angle theorem in detail
A direct consequence: any two inscribed angles that intercept the same arc are equal to each other, regardless of where their vertices sit on the circle. This is one of the most-used facts in circle geometry problems.
The auxiliary-line move
When you're stuck, the single most useful move in angle chasing is drawing an auxiliary line — often a line parallel to one already in the diagram, through a point that lets you split a compound angle into two pieces you already know how to find.
Worked example
Problem: In the diagram, line is parallel to line . A ray from a point between the two lines makes a angle down to and a angle up to . Find the angle between the two rays at that point.
Solution: Draw an auxiliary line through the point, parallel to both and . By alternate interior angles, this splits the angle between the two given rays into two pieces equal to and respectively. So the total angle is.
Practice problems
1.A triangle has two angles measuring and . Find the third angle.
2.An exterior angle of a triangle measures . If one of the remote interior angles is , find the other remote interior angle.
3.An inscribed angle intercepts an arc of . What is the inscribed angle?
4.A cyclic quadrilateral has one angle of . Find the opposite angle.