Congruence and Similarity in Triangles

Converse of Triangle Proportionality Theorem

If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

Triangle Proportionality Theorem

If a line parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides these two sides proportionally.

Hypotenuse-Leg Congruence Theorem (HL)

If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent.

Side-Side-Side Similarity Theorem (SSS)

If the lengths of the corresponding sides of two triangles are proportional, then the triangles must be similar.

Side-Side-Side Congruence Postulate (SSS)

If three sides of one triangle are congruent to three sides of another triangle, then these two triangles are congruent.

Angle-Side-Angle Congruence Postulate (ASA)

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

Angle-Angle-Side Congruence Postulate (AAS)

If two angles and the non-included side one triangle are congruent to two angles and the non-included side of another triangle, then these two triangles are congruent.

Side-Angle-Side Congruence Postulate (SAS)

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then these two triangles are congruent.

Angle-Angle Similarity Postulate (AA)

Two triangles are similar if they have two corresponding angles that are congruent or equal in measure.

Side-Angle-Side Similarity Theorem (SAS)

When two triangles have corresponding angles that are congruent and corresponding sides are proportional, then the triangles are similar.