Congruence
Congruence
The Congruence domain tests whether you can justify that two figures are exactly the same size and shape using rigid motions and triangle congruence criteria. On the Regents Geometry exam, expect transformation descriptions, proofs, constructions, and questions about isosceles triangles and parallel lines.
Core rules
- A congruence is a sequence of rigid motions: translations, rotations, and reflections. These preserve distance and angle measure; reflections reverse orientation.
- Two triangles are congruent by SSS, SAS, ASA, AAS, or HL (right triangles only). SSA and AAA do not guarantee congruence.
- Once congruence is established, corresponding parts are congruent (CPCTC). The order of letters in a congruence statement tells you which parts correspond.
- Isosceles triangle theorem: if two sides are congruent, the angles opposite them are congruent; the converse is also true.
- Constructions with compass and straightedge can copy segments and angles and construct perpendicular bisectors and angle bisectors, which support congruence arguments.
Worked example
In quadrilateral ABCD, AB is parallel to CD and AB is congruent to CD. Prove AD is parallel to BC. Draw diagonal AC. Because AB is parallel to CD, alternate interior angles BAC and DCA are congruent. AC is congruent to CA by the reflexive property. With AB congruent to CD, triangles ABC and CDA are congruent by SAS. Therefore angles BCA and DAC are congruent by CPCTC. These are alternate interior angles formed by transversal AC with lines AD and BC, so AD is parallel to BC.
Traps
Do not use SSA as a congruence shortcut or AAA as proof of congruence. Do not write CPCTC before proving the triangles congruent. Match vertices carefully: AB congruent to DE does not mean AB is congruent to EF. Never assume markings that are not given, and remember that a reflection is a rigid motion even though it reverses orientation.