A triangle is a polygon with three edges and three vertices. It is the simplest two-dimensional closed figure.

The sum of the interior angles of any triangle in Euclidean geometry is always \(180^\circ\) (or \(\pi\) radians).

• Equilateral Triangle: All three sides are equal in length, and all three angles are \(60^\circ\).
• Isosceles Triangle: Two sides are equal in length, and the angles opposite these sides are also equal.
• Scalene Triangle: All three sides have different lengths, and all three angles have different measures.

• Acute Triangle: All three interior angles are acute (less than \(90^\circ\)).
• Right Triangle: One interior angle is exactly \(90^\circ\). The side opposite the right angle is called the hypotenuse, and the other two sides are legs.
• Obtuse Triangle: One interior angle is obtuse (greater than \(90^\circ\)).

• Perimeter: The sum of the lengths of its three sides.
• Area: Commonly calculated as \(\frac{1}{2} \times \text{base} \times \text{height}\). For any triangle, also by Heron's formula if all side lengths are known, or \(\frac{1}{2}ab\sin C\).
• Pythagorean Theorem: For a right triangle with legs \(a, b\) and hypotenuse \(c\), \(a^2 + b^2 = c^2\).
• Trigonometric Ratios: In a right triangle, sine, cosine, and tangent relate angles and side lengths (SOH CAH TOA).
• Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
• Congruence: Triangles are congruent if they have the same size and shape (e.g., SSS, SAS, ASA, AAS criteria).
• Similarity: Triangles are similar if they have the same shape but possibly different sizes (e.g., AA, SSS, SAS criteria for similarity).

• Altitude: A line segment from a vertex perpendicular to the opposite side (or its extension). The altitudes intersect at the orthocenter.
• Median: A line segment from a vertex to the midpoint of the opposite side. The medians intersect at the centroid (center of mass).
• Angle Bisector: A line segment that bisects an angle of the triangle. The angle bisectors intersect at the incenter (center of the inscribed circle).
• Perpendicular Bisector: A line segment that bisects a side and is perpendicular to it. The perpendicular bisectors intersect at the circumcenter (center of the circumscribed circle).

Triangles are widely applied in:

• Architecture and Engineering: For structural stability due to their rigid shape.
• Cartography and Surveying: Triangulation for mapping and determining distances.
• Physics: Vector resolution and force analysis.
• Computer Graphics: Building meshes for 3D models.
• Navigation: Estimating positions using triangulation.