Geometry • Topic 1

Angle Sum

The angle sum properties are among the most fundamental facts in Euclidean geometry. They form the basis for angle chasing, a powerful technique in olympiad geometry.

Triangle Angle Sum

Theorem (Triangle Angle Sum)

The sum of the interior angles of a triangle is

180°

∠ A + ∠ B + ∠ C = 180°

Proof. Draw a line through vertex $A$ parallel to $BC$ . By alternate interior angles with the parallel line:

The angle on one side equals $∠ B$
The angle on the other side equals $∠ C$

These three angles form a straight line, so

∠ A + ∠ B + ∠ C = 180°

. ∎

Exterior Angle Theorem

Theorem (Exterior Angle Theorem)

An exterior angle of a triangle equals the sum of the two non-adjacent interior angles.

Remark

If we extend side

BC

beyond

C

, the exterior angle at

C

equals

∠ A + ∠ B

This follows directly from the angle sum: if the interior angle at $C$ is $γ$ , then the exterior angle is $180° - γ = ∠ A + ∠ B$ .

Polygon Angle Sum

Theorem (Polygon Angle Sum)

For a convex polygon with

n

sides:

Sum of interior angles = (n - 2) \times 180°

Polygon	Sides	Angle Sum

Triangle	3	$180°$
Quadrilateral	4	$360°$
Pentagon	5	$540°$
Hexagon	6	$720°$

Proof. Divide the $n$ -gon into $(n - 2)$ triangles by drawing diagonals from one vertex. Each triangle contributes $180°$ . ∎

Applications in Olympiad Problems

Angle Chasing

Most geometry problems involve finding unknown angles. The key is to:

Mark all known angles
Use angle sum in triangles
Use properties of special configurations (isosceles triangles, cyclic quadrilaterals, etc.)

Example

In triangle

A BC

∠ A = 50°

and the angle bisector from

B

meets

A C

D

. If

∠ A D B = 80°

, find

∠ C

Proof (Solution). Let $∠ A B D = ∠ D BC = β$ (angle bisector).

In $△ A B D$ : $50° + β + 80° = 180°$ , so $β = 50°$ .

In $△ A BC$ : $50° + 2 β + ∠ C = 180°$

Therefore $∠ C = 180° - 50° - 100° = 30°$ . ∎

Directed Angles

Tip (Advanced Technique)

For more advanced problems, we use directed angles modulo

180°

. This elegantly handles configuration issues and simplifies proofs involving cyclic quadrilaterals.

Key Lemmas

Lemma (Isosceles Triangle)

A B = A C

, then

∠ B = ∠ C = \frac{180° - ∠ A}{2}

Lemma (Right Triangle)

∠ C = 90°

, then

∠ A + ∠ B = 90°

Lemma (Inscribed Angle)

An inscribed angle is half the central angle subtending the same arc.

Practice Problems

Exercise (Problem 1)

In triangle

A BC

∠ A = 40°

. The altitude from

B

meets

A C

H

, and the altitude from

C

meets

A B

K

. Find

∠ B H C

and

∠ HK C

Exercise (Problem 2)

In a convex pentagon, four of the angles are equal. If the fifth angle is

60°

, find the measure of each of the equal angles.

Exercise (Problem 3 (Classic))

Prove that in any triangle, the sum of any two angles is greater than

90°

if and only if the triangle is acute.