Angle Sum
The angle sum properties are among the most fundamental facts in Euclidean geometry. [cite_start]They form the basis for "angle chasing," a powerful technique in olympiad geometry[cite: 50, 51].
Triangle Angle Sum
Theorem (Triangle Angle Sum)
The sum of the interior angles of a triangle is .
[cite_start][cite: 52, 53]
Proof. Draw a line through vertex parallel to . By alternate interior angles with the parallel line:
- The angle on one side equals
- The angle on the other side equals
Exterior Angle Theorem
Theorem (Exterior Angle Theorem)
[cite_start]An exterior angle of a triangle equals the sum of the two non-adjacent interior angles[cite: 56].
Remark
If we extend side beyond , the exterior angle at equals .
[cite_start]This follows directly from the angle sum: if the interior angle at is , then the exterior angle is [cite: 57].
Polygon Angle Sum
Theorem (Polygon Angle Sum)
For a convex polygon with sides:
[cite_start][cite: 58]
| Polygon | Sides | Angle Sum |
|---|
| Triangle | 3 | |
|---|---|---|
| Quadrilateral | 4 | |
| Pentagon | 5 | |
| Hexagon | 6 |
Proof. Divide the -gon into triangles by drawing diagonals from one vertex. [cite_start]Each triangle contributes [cite: 61]. ∎
Practice Problems
Exercise (Problem 1)
In triangle , . The altitude from meets at , and the altitude from meets at . [cite_start]Find and [cite: 70, 71, 72].
Exercise (Problem 2)
In a convex pentagon, four of the angles are equal. [cite_start]If the fifth angle is , find the measure of each of the equal angles[cite: 73].
Exercise (Problem 3 (Classic))
[cite_start]Prove that in any triangle, the sum of any two angles is greater than if and only if the triangle is acute[cite: 75].