Congruent Triangles - Quick Reference

Congruent Triangles

• Definition: Two triangles are congruent if one can be superimposed on the other (rotation/reflection allowed).
• Notation: $\triangle ABC \cong \triangle DEF$
• Corresponding parts are equal.

Congruence Tests

• SSS: three pairs of corresponding sides are equal
  • Pattern: $AB = DE,\; BC = EF,\; CA = FD$
• SAS: two sides and the included angle are equal
  • Pattern: $AB = DE,\; \angle A = \angle D\; (included angle),\; AC = DF$
• ASA: two angles and the included side are equal
  • Pattern: $\angle A = \angle D,\; \angle C = \angle F,\; AC = DF$
• AAS: two angles and a non-included side are equal
  • Pattern: $\angle A = \angle D,\; \angle B = \angle E,\; AB = DE$
• RHS: right-angled triangles; hypotenuse and one corresponding side are equal
  • Pattern: if both triangles are right with $\angle C = 90^{\circ},\; \angle F = 90^{\circ},\; BC = EF$ then $\triangle ABC \cong \triangle DEF$

Quick Reference: When to Use

• SSS: three sides known
• SAS: two sides and the included angle known
• ASA: two angles and the included side known
• AAS: two angles and a non-included side known
• RHS: both triangles right-angled with hypotenuse and a leg known

Real-World Use

• Construction: congruent triangles reinforce structures for strength and stability under load

Patterns (Examples)

• SSS pattern: $\triangle ABC \cong \triangle DEF$ if $AB = DE,\; BC = EF,\; CA = FD$
• SAS pattern: $\triangle ABC \cong \triangle DEF$ if $AB = DE,\; AC = DF,\; \angle A = \angle D$
• ASA pattern: $\triangle ABC \cong \triangle DEF$ if $\angle A = \angle D,\; \angle C = \angle F,\; AC = DF$
• AAS pattern: $\triangle ABC \cong \triangle DEF$ if $\angle A = \angle D,\; \angle B = \angle E,\; AB = DE$
• RHS pattern: right triangles with congruent hypotenuse and a leg are congruent