Properties of Triangles

Properties of Triangles

• Angles:
  • Sum of interior angles: 180°
  • Exterior angle: equal to the sum of the two non-adjacent interior angles
  • Angle sum property: the sum of the angles of a triangle is constant
• Sides:
  • Sum of any two sides: greater than the third side
  • Difference between any two sides: less than the third side
• Congruent Triangles:
  • SSS (Side-Side-Side): if three sides are equal, the triangles are congruent
  • SAS (Side-Angle-Side): if two sides and the included angle are equal, the triangles are congruent
  • ASA (Angle-Side-Angle): if two angles and the included side are equal, the triangles are congruent
  • AAS (Angle-Angle-Side): if two angles and a non-included side are equal, the triangles are congruent
• Similar Triangles:
  • AA (Angle-Angle): if two angles are equal, the triangles are similar
  • SSS (Side-Side-Side): if three sides are proportional, the triangles are similar
• Right Triangles:
  • Pythagorean theorem: a^2 + b^2 = c^2, where c is the hypotenuse
  • Trigonometric ratios: sine, cosine, and tangent
• Triangle Inequality:
  • The length of any side is less than the sum of the lengths of the other two sides
  • The length of any side is greater than the positive difference of the lengths of the other two sides

Angles in Triangles

• Sum of interior angles in a triangle is 180°
• An exterior angle is equal to the sum of the two non-adjacent interior angles

Properties of Triangle Sides

• The sum of any two sides of a triangle is greater than the third side
• The difference between any two sides of a triangle is less than the third side

Congruent Triangles

• If three sides of two triangles are equal, the triangles are congruent (SSS)
• If two sides and the included angle of two triangles are equal, the triangles are congruent (SAS)
• If two angles and the included side of two triangles are equal, the triangles are congruent (ASA)
• If two angles and a non-included side of two triangles are equal, the triangles are congruent (AAS)

Similar Triangles

• If two angles of two triangles are equal, the triangles are similar (AA)
• If three sides of two triangles are proportional, the triangles are similar (SSS)

Right Triangles

• The Pythagorean theorem states that a^2 + b^2 = c^2, where c is the hypotenuse
• Trigonometric ratios in right triangles include sine, cosine, and tangent

Triangle Inequality Theorem

• The length of any side of a triangle is less than the sum of the lengths of the other two sides
• The length of any side of a triangle is greater than the positive difference of the lengths of the other two sides

Learn about the properties of triangles, including angles, sides, and congruent triangles. Understand the sum of interior angles, exterior angle, and angle sum property. Explore the rules for sides and congruent triangles.