Definition of Congruence of Triangles
Two triangles are congruent if all corresponding sides and corresponding angles are equal.
Congruence Postulates
1. Side-Side-Side (SSS) Postulate
If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.
Example:
For \(\triangle ABC\) and \(\triangle DEF\):
If \(AB = DE\), \(BC = EF\), and \(AC = DF\), then \(\triangle ABC \cong \triangle DEF\) where \(\cong\) is the symbol of congruence of two triangles.
2. Side-Angle-Side (SAS) Postulate
If two sides and the included angle of one triangle are equal to those of another triangle, the triangles are congruent.
Example:
If \(AB = DE\), \(\angle A = \angle D\), and \(AC = DF\), then \(\triangle ABC \cong \triangle DEF\).
3. Angle-Side-Angle (ASA) Postulate
If two angles and the included side of one triangle are equal to those of another triangle, the triangles are congruent.
Example:
If \(\angle B = \angle E\), \(BC = EF\), and \(\angle C = \angle F\), then \(\triangle ABC \cong \triangle DEF\).
4. Angle-Angle-Side (AAS) Theorem
If two angles and a non-included side of one triangle are equal to those of another triangle, the triangles are congruent.
Example:
If \(\angle A = \angle D\), \(\angle C = \angle F\), and \(AB = DE\), then \(\triangle ABC \cong \triangle DEF\).
5. Hypotenuse-Leg (HL) Theorem for Right Triangles
If the hypotenuse and one leg of a right triangle are equal to those of another right triangle, the triangles are congruent.
Example:
For right triangles \(\triangle ABC\) and \(\triangle DEF\):
If \(AC = DF\) (hypotenuse) and \(BC = EF\) (leg), then \(\triangle ABC \cong \triangle DEF\).
Important Corollaries
CPCTC
Corresponding Parts of Congruent Triangles are Congruent:
If \(\triangle ABC \cong \triangle DEF\), then all corresponding angles and sides are equal.
SSA (Non-Congruence Case)
SSA does not guarantee congruence (ambiguous case). It works only for right triangles (HL).
Counterexample:
Two triangles with \(AB = DE\), \(BC = EF\), and \(\angle A = \angle D\) might not be congruent.