Congruence and Similarity
Congruence questions hinge on the triangle congruence criteria: SSS, SAS, ASA, AAS, and HL (right triangles only). SSA and AAA do not guarantee congruence. Be ready to read a marked diagram and select the criterion that applies, and to supply a missing piece: for example, if two sides and a non-included angle are given, you may need to add a right angle to use HL, or add the included angle to use SAS. Corresponding parts of congruent triangles are congruent (CPCTC) is usually the last step of a proof, not the first.
Similarity rests on AA, SSS similarity, and SAS similarity. Once triangles are similar, corresponding sides are proportional and corresponding angles are congruent; the ratio of perimeters equals the scale factor, while the ratio of areas equals the square of the scale factor, and the ratio of volumes equals its cube. Traps include setting up proportions with mismatched corresponding sides, forgetting to square the scale factor for area, and assuming triangles are similar just because they share an angle (you need two pairs).
Rule of thumb. Write the similarity statement first, matching vertices in order (triangle ABC ~ triangle DEF). Then every proportion you write follows directly from that letter order: AB/DE = BC/EF = AC/DF. If a proportion looks wrong, check it against the statement.
Worked example. Two similar triangles have a scale factor of 3:5. If the smaller triangle has area 18, the larger area is 18 times (5/3)^2 = 18 times 25/9 = 50.