Triangle Congruence and Similarity

Key Concepts

• Triangle Congruence: Two triangles are congruent if all corresponding sides and angles are equal.
• Triangle Similarity: Two triangles are similar if they have the same shape but not necessarily the same size.

Congruence Postulates

Side-Side-Side (SSS)

• If all three sides of one triangle are equal to all three sides of another triangle, the triangles are congruent.

Angle-Angle-Angle (AAA)

• Not for Congruence: Having all three corresponding angles equal does not imply congruence.
• Implies similarity instead, meaning the triangles have the same shape but can be scaled versions of each other.

Side-Angle-Side (SAS)

• If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.
• The angle must be between the two sides.

Angle-Side-Angle (ASA)

• If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent.
• The side must be between the two angles.

Angle-Angle-Side (AAS)

• If two angles and any side of one triangle are equal to two angles and any side of another triangle, the triangles are congruent.

Non-Congruence and Non-Similarity Postulates

Side-Side-Angle (SSA)

• Does not necessarily imply congruence or similarity.
• Different configurations can result in triangles that are not congruent, even if two sides and a non-included angle are the same.

Summary

• Congruent triangles are also similar, but similar triangles are not necessarily congruent.
• Understanding these postulates helps in determining whether triangles are congruent or similar using minimal information.