Congruent Triangles

Definition

Two triangles are congruent ($\cong$) if they have exactly the same shape and size. Formally, this means all six pairs of corresponding parts (three sides and three angles) are congruent.

• How to read: “The triangle A is congruent to triangle B.”
• Meaning: Same shape and size, superimposable—all corresponding sides and angles match.

Why It Matters

It provides the minimum set of conditions needed to prove that two objects are physically identical, enabling precise engineering and design.

Core Concepts

• Sufficient Conditions for Congruence:
  • SSS: Three pairs of congruent sides.
  • SAS: Two sides and the included angle.
  • ASA: Two angles and the included side.
  • SAA / AAS: Two angles and a non-included side.
  • HL: Hypotenuse and one leg (right triangles only).
• Identifying Corresponding Parts:
  • Corresponding sides lie opposite congruent angles.
  • Corresponding angles lie opposite congruent sides.
• Proof Strategy:
  1. Identify the triangles to be proven congruent.
  2. Locate three pairs of congruent parts (Givens, vertical angles, common parts, bisectors).
  3. Apply the appropriate principle (SSS, SAS, ASA, SAA, HL).
  4. Use CPCTC to prove the specific target parts are congruent.

Connected Concepts

• CPCTC
• Similar Triangles
• Symmetry
• Right Triangles