Chapter 6: The Triangle and its Properties

Introduction to The Triangle and its Properties

The Triangle and its Properties, A triangle is one of the simplest shapes in geometry, formed by three straight lines that meet at three points called vertices. It has three sides and three angles. In this chapter, we explore the properties of triangles, types of triangles, and some special features like medians, altitudes, and angle sums.

1. Basic Definitions

• Triangle: A closed figure with three sides, three angles, and three vertices. For example, if A, B, and C are three points, the figure formed by joining AB, BC, and CA is a triangle, written as ∆ABC.
• Sides: The three line segments that form the triangle (e.g., AB, BC, CA).
• Vertices: The points where the sides meet (e.g., A, B, C).
• Angles: The three angles formed at the vertices (e.g., ∠A, ∠B, ∠C).

2. Types of Triangles

Triangles can be classified based on their sides or angles.

Based on Sides:

1. Equilateral Triangle:
   • All three sides are equal (AB = BC = CA).
   • All three angles are equal, and each angle measures 60°.
   • Example: A triangle with sides 5 cm, 5 cm, and 5 cm.
2. Isosceles Triangle:
   • Two sides are equal (e.g., AB = AC).
   • The angles opposite the equal sides are also equal (∠B = ∠C).
   • Example: A triangle with sides 4 cm, 4 cm, and 6 cm.
3. Scalene Triangle:
   • No sides are equal (AB ≠ BC ≠ CA).
   • No angles are equal.
   • Example: A triangle with sides 3 cm, 4 cm, and 5 cm.

Based on Angles:

1. Acute-Angled Triangle:
   • All three angles are less than 90°.
   • Example: A triangle with angles 50°, 60°, and 70°.
2. Right-Angled Triangle:
   • One angle is exactly 90°.
   • The side opposite the right angle is called the hypotenuse, the longest side.
   • Example: A triangle with angles 90°, 30°, and 60°.
3. Obtuse-Angled Triangle:
   • One angle is greater than 90°.
   • Example: A triangle with angles 110°, 40°, and 30°.

3. Angle Sum Property of a Triangle

One of the most important properties of a triangle is that the sum of its three interior angles is always 180°.

• Statement: ∠A + ∠B + ∠C = 180°.
• Proof: Imagine drawing a line parallel to one side of the triangle through a vertex. Using the properties of parallel lines and alternate angles, it can be shown that the sum of the triangle’s angles equals 180°.
• Example: In ∆ABC, if ∠A = 40° and ∠B = 70°, then ∠C = 180° – (40° + 70°) = 70°.

4. Exterior Angle Property

An exterior angle of a triangle is formed when one side of the triangle is extended.

• Statement: The measure of an exterior angle is equal to the sum of the two opposite interior angles.
• Example: In ∆ABC, if ∠ACD is an exterior angle at C (formed by extending BC), then ∠ACD = ∠A + ∠B.
• Verification: If ∠A = 50° and ∠B = 60°, then ∠ACD = 50° + 60° = 110°.

5. Medians of a Triangle

A median is a line segment that joins a vertex of a triangle to the midpoint of the opposite side.

• Key Points:
  • Every triangle has three medians (one from each vertex).
  • The medians intersect at a point called the centroid, denoted by G.
  • The centroid divides each median in the ratio 2:1 (the part from the vertex to the centroid is twice the part from the centroid to the midpoint).
• Example: In ∆ABC, if D is the midpoint of BC, then AD is a median. Similarly, BE (E is the midpoint of AC) and CF (F is the midpoint of AB) are the other medians.

6. Altitudes of a Triangle

An altitude is a perpendicular line segment from a vertex to the line containing the opposite side (or its extension).

• Key Points:
  • Every triangle has three altitudes.
  • The altitudes intersect at a point called the orthocentre, denoted by H.
  • In a right-angled triangle, the orthocentre is at the vertex of the right angle.
• Example: In ∆ABC, if AD is perpendicular to BC, then AD is an altitude. Similarly, BE ⊥ AC and CF ⊥ AB are altitudes.

7. Properties of Special Triangles

• Equilateral Triangle:
  • All sides are equal, and all angles are 60°.
  • The centroid, orthocentre, and other special points coincide at the same point.
• Right-Angled Triangle:
  • Pythagoras Theorem applies: In ∆ABC with ∠B = 90°, AC² = AB² + BC² (AC is the hypotenuse).
  • Example: If AB = 3 cm, BC = 4 cm, then AC = √(3² + 4²) = √(9 + 16) = 5 cm.

8. Triangle Inequality Property

The sum of the lengths of any two sides of a triangle is always greater than the length of the third side.

• Statement: In ∆ABC, AB + BC > AC, BC + CA > AB, and CA + AB > BC.
• Example: If AB = 3 cm, BC = 4 cm, and AC = 5 cm, check:
  • 3 + 4 > 5 (7 > 5, true).
  • 4 + 5 > 3 (9 > 3, true).
  • 5 + 3 > 4 (8 > 4, true).
• This property ensures that a triangle can be formed with the given side lengths.

9. Examples and Solved Problems

1. Finding the Third Angle:
   • In ∆PQR, ∠P = 45°, ∠Q = 65°. Find ∠R.
   • Solution: ∠R = 180° – (45° + 65°) = 180° – 110° = 70°.
2. Exterior Angle:
   • In ∆XYZ, ∠X = 50°, ∠Y = 60°. Find the exterior angle at Z.
   • Solution: Exterior angle at Z = ∠X + ∠Y = 50° + 60° = 110°.
3. Pythagoras Theorem:
   • In a right-angled triangle with legs 6 cm and 8 cm, find the hypotenuse.
   • Solution: Hypotenuse = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.

10. Key Points to Remember

• Sum of angles in a triangle is 180°.
• Exterior angle equals the sum of the two opposite interior angles.
• Medians meet at the centroid, altitudes meet at the orthocentre.
• Use the triangle inequality to check if a triangle is possible with given sides.
• Pythagoras Theorem is specific to right-angled triangles.

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Chapter 7: Congruence of Triangles

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