Triangle Inequality

• Exterior Angle Inequality Theorem


  The measure of any exterior angle of a triangle is greater than both of the non-adjacent interior angles

  
• Exterior angle of a triangle

  1. Angle formed between any side of the triangle and an external extension of the adjacent side

  2. The remote exterior angles are the angles in the interior of the triangle that do not form a linear pair with the corresponding exterior angle

  
• An exterior angle of a triangle is greater than either of the non-adjacent interior angles

  
• Triangle Inequality Theorem


  In a triangle, the sum of the lengths of any two sides is greater than the length of the third side
• Can a triangle be constructed with sides of lengths 12cm, 24cm, and 18cm?

  • Yes
  • No

  
• If one side of a triangle is longer than the second, then the angle opposite the longer side is larger than the side opposite the second side
• If one angle of a triangle is larger than the second angle, then the side opposite the larger angle is longer than the side opposite the second

  
• The sum of the measures of the interior angles of a triangle is 180°

  
• If 4cm, 8cm, and 2cm are the measure of three line segments, can it be used to draw a triangle?

  
• Unequal Sides Theorem


  If one side of a triangle is longer than the second, then the angle opposite the longer side is greater than the side opposite the second side
• Unequal Angles Theorem


  If one angle of a triangle is larger than the second angle, then the side opposite the larger angle is longer than the side opposite the second

  
• The theorem which states that the measure of any exterior angle of a triangle is greater than both of the non-adjacent interior angles is the Exterior Angle Inequality theorem

  
• The shortest side of ΔMOB is BM
• The theorem that stated that in a triangle, the sum of the lengths of any two sides is greater than the length of the third side is the Triangle Inequality Theorem
• In ΔTRU, TR = 9 cm, RU = 10 cm, and TU = 11 cm. The angles in order from least to greatest measure are m∠R, m∠T, m∠U
• The sides of ΔKEY in order from least to greatest measure are EK, YK, EY
• The size of an angle

  The length of the side opposite it
• The largest angle of a triangle

  The side opposite it

  
• The smallest angle of a triangle

  The side opposite it

  
• Scissors, fliers, and many other tools are hinged and are often use in our daily lives
• Hinge theorem and its converse are applicable in real life
• A bisector divides an angle into two congruent angles
• The exterior angle of a triangle is equal to the sum of the two nonadjacent interior angles of the triangle

  
• The whole is greater than its parts
• In a triangle, the longest side is opposite the largest angle

  
• In a triangle, the sum of the lengths of any two sides is greater than the length of the third side
• Workers, designers, planners, engineers and architects make use of triangle in making their plans and designs, they used mathematical concept and skills in triangle inequalities to help and to justify all the triangular details in their works