Geometry • Topic 5

Angle Chasing

Angle chasing is the art of finding unknown angles by successively applying geometric properties. It is the primary method for solving problems involving cyclic quadrilaterals, tangents, and special triangle centers.

The Toolkit

To be effective at angle chasing, you must instantly recognize these configurations:

Triangles: Sum is $18 0^{\circ}$ . [cite_start]Exterior angle equals sum of two opposite interior angles. [cite: 52, 56]
[cite_start]Isosceles Triangles: Base angles are equal. [cite: 68]
Parallel Lines: Alternate interior ("Z") angles are equal.
Circle Properties:

[cite_start]Angles subtended by the same arc are equal. [cite: 70]

Tangent-Chord angle equals the angle in the alternate segment.

Center angle is double the inscribed angle.

Strategy: Directed Angles

In complex configurations, the direction of angles matters. Directed angles (angles mod

18 0^{\circ}

) eliminate the need for separate cases (like "point

P

is inside vs outside").

Notation: $∠ (A B, C D)$ denotes the angle required to rotate line $A B$ counterclockwise to become parallel to $C D$ .
Property: Four points $A, B, C, D$ are concyclic if and only if $∠ (A C, CB) = ∠ (A D, D B)$ .

Algebraic Angle Chasing

Sometimes you cannot find numerical values. Instead, assign variables (

α, β

) to key angles and express others in terms of them.

Example

△ A BC

A B = A C

. Point

D

is on

A C

such that

B D

bisects

∠ B

. If

BC = B D + A D

, find

∠ A

Proof. Let $∠ A = α$ . Since $A B = A C$ , $∠ B = ∠ C = \frac{180 - α}{2} = 90 - α /2$ . Since $B D$ bisects $∠ B$ , $∠ D BC = ∠ A B D = 45 - α /4$ . In $△ BC D$ , apply sine rule or trace lengths... actually, this is the classic "Langley's Adventitious Angles" type problem, often solvable by pure angle chasing if auxiliary lines are drawn. Solution hint: Construct a point $E$ on $BC$ such that $BE = B D$ ... ∎

Practice Problems

Exercise (Problem 1)

In acute

△ A BC

, altitudes

A D

and

BE

meet at

H

. If

∠ A CB = 4 5^{\circ}

, find

∠ A H B

Exercise (Problem 2)

Two circles intersect at

A

and

B

. A line through

A

meets the circles again at

C

and

D

. A line through

B

meets the circles again at

E

and

F

. Prove that

CE ∥ D F

Exercise (Problem 3)

△ A BC

, the angle bisector of

A

meets the circumcircle at

D

. Prove that the center of the incircle lies on

A D