All About Triangles

Triangles

A triangle is a two-dimensional shape with three sides, three angles, and three vertices (the points where the sides connect). Because it has three or more sides, a triangle is also considered a polygon, so it belongs in the same shape family as the square, the rectangle, and even the pentagon (which has five sides!).

Triangle Properties

There are a few properties or rules that a shape must have in order to be considered a triangle

  • A triangle must have three sides
  • The sum of all interior angles must equal \( 180^o \)
  • The sum of the lengths of any two sides is always greater than the length of the third side
  • The difference between the lengths of any two sides of a triangle is less than the length of the third side
  • The shortest side is always opposite the smallest interior angle. Similarly, the longest side is always opposite the largest interior angle.

Types of Triangles

Trainagles can be categorised using either its interior angles or the length of its sides – or both.

Type Figure  
Triangles based on angles
Acute triangles a b c A B C
  • All angles are of different and are \( < 90^o \)
  • All three sides are of different length
Obtuse triangles a b c A B C
  • One of the angle is \( > 90^o \)
  • All three sides are of different length
Right triangles a b c A B C
  • One of the angles is equal to \( 90^o \)
  • Sum of the other two angles is equal to \( 90^o \)
  • Side opposite to right angle is the longest side, know as hypotenuse
Triangles based on sides
Scalene triangles a b c A B C
  • All three sides are of different lengths \( a \ne b \ne c \)
  • All three angles are different from each other
Equilateral triangles a b c A B C
  • All three sides are equal to each other \( a = b = c \)
  • All three internal angles are equal to \( 60^o \) each
Isosceles triangles a b c A B C
  • Two sides are equal to each other \( a = b \)
  • Angles opposite to equal sides are equal

Solving Triangles

"Solving" means finding missing sides and angles of the triangle. This can be achieved by using Laws of Sines and Cosines.

Type Figure  
AAS (Two Angles and a Side not between) a b c A B C
  • Find the missing angle. Sum of all 3 angles is always \( = 180^o\)
  • Use Law of Sines to find each of the other two sides
ASA (Two Angles and a Side between) a b c A B C
  • Find the missing angle. Sum of all 3 angles is always \( = 180^o\)
  • Use Law of Sines to find each of the other two sides
SAS (Two Sides and an Angle between) a b c A B C
  • Use Law of Cosines to calculate the unknown side
  • Use The Law of Sines to find the smaller of the other two angles
  • Find the missing angle. Sum of all 3 angles is always \( = 180^o\)
SSA (Two Sides and an Angle not between) a b c A B C
  • Use Law of Sines to calculate one of the other two angles
  • Find the missing angle. Sum of all 3 angles is always \( = 180^o\)
  • Use Law of Sines to find the unknown side
SSS (Three Sides) a b c A B C
  • Use Law of Cosines to calculate one of the angles
  • Use Law of Cosines to find another angle
  • Find the last angle. Sum of all 3 angles is always \( = 180^o\)