Triangles are fundamental geometric shapes with several properties that mathematicians and scientists study. Here are some key properties of triangles:

  1. Three Sides:-  A triangle is a polygon with three sides. Each side is a line segment that connects two vertices.
  2. Three Angles:-  A triangle has three internal angles formed where its sides intersect. The sum of the interior angles of a triangle is always 180∘ (or π radians).
  3. Types Based on Sides:-  Equilateral Triangle:- All three sides are of equal length, and all three angles are equal, each measuring 60∘. Isosceles Triangle:- Two sides are of equal length. Scalene Triangle:- No sides are of equal length.
  4. Types Based on Angles:-  Acute Triangle:- All interior angles are less than 90∘. Right Triangle:- One interior angle is 90∘. The side opposite the right angle is the hypotenuse. Obtuse Triangle:- One interior angle is greater than 90∘.
  5. 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. Mathematically, if a, b, and c are the lengths of the sides of a triangle, then:- (a + b > c), (a + c > b), (b + c > a)
  6. Altitudes, Medians, and Angle Bisectors:- Altitudes:- Lines drawn from each vertex perpendicular to the opposite side. Medians:- Lines drawn from each vertex to the midpoint of the opposite side. Angle Bisectors:- Lines drawn from each vertex bisecting the opposite angle.

   7. Area Formulas:- 

     (i) Using base and height:-

                      Area = 1/2 ​× base × height

     (ii) Using Heron's formula for the area of a triangle with sides a, b, and c:-

                      Area = √s(s−a) (s−b) (s−c)​

   where s is the semiperimeter of the triangle defined as s = (a+b+c)/2​.

   8. Similarity and Congruence:- Triangles can be similar (same shape, different size) or congruent (same shape and size).

Understanding these properties helps in solving geometric problems involving triangles and applying triangle properties in various fields of science, engineering, and mathematics.