Triangle and its Properties - Chapter 6

Introduction

• Chapter name: Triangle and its Properties
• Purpose: Understand concepts for easy exercise completion.

Definition of Triangle

• A triangle is a closed figure with three line segments.
• Examples of triangles: Various shapes can form triangles if three line segments join.

Parts of a Triangle

Sides

• Example Triangle: PQR
  • Sides: PQ, QR, PR

Angles

• Angles in Triangle PQR:
  • Angle P, Angle Q, Angle R
  • Notation: Angle P can be written as Angle QPR or Angle RQP.

Vertices

• Definition: Corner points of the triangle.
• Vertices of Triangle PQR: P, Q, R.

Key Concepts

Median

• Definition: A line connecting a vertex to the midpoint of the opposite side.
• Example:
  • Triangle ABC, median from vertex B to midpoint D of AC.
  • Medians in triangle: BD, AD, CD.

Altitude

• Definition: A line from a vertex perpendicular to the opposite side.
• Example:
  • Triangle ABC, altitude from vertex A to side BC, forming a right angle.
  • Not to be confused with median, as altitude does not necessarily pass through the midpoint.

Exterior Angles

• Definition: Angles formed outside the triangle when sides are extended.
• Property: An exterior angle is equal to the sum of the two opposite interior angles.
  • Example: If Angle 1 and Angle 2 are interior, then the exterior angle equals Angle 1 + Angle 2.

Angles and Properties of Triangle

• Property: The sum of the measures of the three angles in a triangle is 180 degrees.
  • Example: If two angles are 60° and 70°, the third angle is 180° - (60° + 70°) = 50°.

Types of Triangles

Equilateral Triangle

• Definition: All three sides and angles are equal.
  • Each angle measures 60°.

Isosceles Triangle

• Definition: Any two sides are equal.
• Property: If two sides are equal, the angles opposite those sides are also equal.

Triangle Properties

• Triangle Inequality Theorem: The sum of any two sides must be greater than the third side.
  • Example: For triangle ABC, AB + BC > AC, AB + AC > BC, BC + AC > AB.

Right Angle Triangle and Pythagorean Theorem

Right Angle Triangle

• Definition: Triangle with one angle measuring 90°.
• Hypotenuse: The side opposite the right angle.
• Legs: The other two sides.

Pythagorean Theorem

• Definition: In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
  • Formula: c² = a² + b²
  • Example: For triangle with legs of 5 cm and 12 cm, find hypotenuse.
    • 5² + 12² = 25 + 144 = 169
    • Hypotenuse = √169 = 13 cm.

Conclusion

• Summary: All concepts discussed will aid in understanding exercises.
• Next chapter will be introduced in future presentations.