Chapter 6: The Triangle and Its Properties

6.1 Introduction to Triangles

• Definition: A triangle is a simple closed curve made of three line segments.
  • Elements:
    • Sides: AB, BC, CA
    • Angles: BAC, ABC, BCA
    • Vertices: A, B, C
• Classification:
  • By Sides: Scalene, Isosceles, Equilateral
  • By Angles: Acute-angled, Obtuse-angled, Right-angled

6.2 Medians of a Triangle

• A median of a triangle connects a vertex to the midpoint of the opposite side.
• Properties:
  • Each triangle has 3 medians.
  • Medians always lie within the triangle.

6.3 Altitudes of a Triangle

• An altitude is a perpendicular from a vertex to the line containing the opposite side.
• Properties:
  • Each triangle has 3 altitudes.
  • Altitudes may lie outside the triangle for obtuse triangles.

6.4 Exterior Angle of a Triangle and Its Property

• An exterior angle is formed when a side of a triangle is extended.
• Property: The measure of an exterior angle is equal to the sum of the measures of its two interior opposite angles.

6.5 Angle Sum Property of a Triangle

• The sum of the angles in any triangle is always 180°.
• This can be demonstrated through various activities involving cutting and rearranging angles.

6.6 Two Special Triangles: Equilateral and Isosceles

• Equilateral Triangle:
  • All sides and angles are equal (each angle = 60°).
• Isosceles Triangle:
  • Two sides are of equal length.
  • Angles opposite the equal sides are equal.

6.7 Sum of the Lengths of Two Sides of a Triangle

• The sum of the lengths of any two sides of a triangle is greater than the third side.
• This helps in determining if three line segments can form a triangle.

6.8 Right-Angled Triangles and Pythagoras Property

• Pythagoras Property: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
• This property is used to verify if a triangle is right-angled.

Key Exercises and Problems

• Use medians and altitudes to solve problems related to the triangle's geometry.
• Apply the exterior angle and angle sum properties in various exercises.
• Solve problems using the Pythagoras theorem to find unknown sides in right-angled triangles.

Summary of Important Properties

1. A triangle’s six elements: three angles and three sides.
2. Medians and altitudes: A triangle has 3 of each, connecting vertices to opposite sides.
3. Exterior angle property: Equals sum of opposite interior angles.
4. Angle sum property: Total of angles in a triangle = 180°.
5. Equilateral and isosceles triangles have specific properties related to side and angle equality.
6. Side length property: Sum of lengths of any two sides > the third side.
7. Pythagoras theorem is applicable to right-angled triangles, confirming their properties.