Chapter 7: Triangles

7.1 Introduction

• A triangle is a closed figure formed by three intersecting lines.
• Components of a triangle: 3 sides, 3 angles, 3 vertices.
• Study of congruence of triangles, rules of congruence, properties, and inequalities.

7.2 Congruence of Triangles

• Congruent figures: identical in shape and size (e.g., identical photographs, ATM cards).
• Congruent circles and squares match completely when placed on top of each other.
• Congruence is used in real-life applications like moulding.
• Example: Congruent triangles: If all sides and angles of one triangle match another, they are congruent.
• Symbolic representation: PQR ≅ ABC implies each side and angle of one corresponds exactly to the other.

7.3 Criteria for Congruence of Triangles

• Four criteria for triangle congruence:
  • SAS (Side-Angle-Side) Rule: Two triangles are congruent if two sides and the included angle of one are equal to those of the other.
  • ASA (Angle-Side-Angle) Rule: Two triangles are congruent if two angles and the included side of one are equal to those of the other.
  • AAS (Angle-Angle-Side) Rule: Two triangles are congruent if two angles and a corresponding side of one are equal to those of the other.
  • SSS (Side-Side-Side) Rule: Two triangles are congruent if all three sides of one are equal to those of the other.
  • RHS (Right angle-Hypotenuse-Side) Rule: In right-angled triangles, if the hypotenuse and one side of one are equal to those of the other, they are congruent.

7.4 Some Properties of a Triangle

• Isosceles triangle: Two sides are equal, and angles opposite these sides are equal.
• The converse: If two angles are equal, then the sides opposite them are equal.

7.5 Some More Criteria for Congruence of Triangles

• Confirmed that equality of three angles is not enough for congruence.
• SSS Rule: Validated through construction and comparison activities.

7.6 Inequalities in a Triangle

• Theorem 7.6: If two sides of a triangle are unequal, the angle opposite the longer side is larger.
• Theorem 7.7: In any triangle, the side opposite the larger angle is longer.
• Theorem 7.8: The sum of any two sides of a triangle is greater than the third side.

7.7 Summary

• Congruent figures: same shape and size.
• Specific congruence rules (SAS, ASA, AAS, SSS, RHS).
• Properties of isosceles and equilateral triangles.
• Inequalities: Relating sides and angles in triangles.

Exercises: Include practical applications and proof exercises related to the discussed theorems and congruence rules.