Three matching parts, chosen correctly, force the whole triangle to match.
Which test?
Two triangles have two pairs of equal angles and the pair of sides between those angles is equal. Which test applies?
Angle, then the included side, then angle — that is ASA. C is AAS, where the side sits outside the two angles. Both are valid tests, and the only difference is where the side lies, so the word "between" is what decides it. Reading the letters in order tells you the arrangement.
△ABC ≅ △DEF means A↔D, B↔E, C↔F, in that order.
Congruent triangles have the same shape AND the same size: every side and every angle equals its partner.
| Corresponding angles | Corresponding sides |
|---|---|
| ∠A = ∠D | AB = DE |
| ∠B = ∠E | BC = EF |
| ∠C = ∠F | AC = DF |
△ABC ≅ △DEF and △ABC ≅ △EDF say different things. Write the letters so that corresponding vertices line up, and every pairing can then be read straight off.
| Test | What you need |
|---|---|
| SSS | all three sides |
| SAS | two sides and the angle BETWEEN them |
| ASA | two angles and the side BETWEEN them |
| AAS | two angles and a side not between them |
| HL | right triangles: hypotenuse and one leg |
Read the letters as a journey around the triangle. In SAS you meet a side, an angle, then a side — so the angle is squeezed between them.
AAA gives similarity, not congruence — a photograph and its enlargement have identical angles. Congruence needs at least one side to fix the size.
two right triangles
hypotenuses equal, one leg equal
Pythagoras fixes the third side
b² = c² − a²
so SSS follows
congruent ✓
HL looks like SSA, which fails in general. It works here only because the right angle pins the third side down through the Pythagorean theorem.
Corresponding Parts of Congruent Triangles are Congruent.
prove △ABC ≅ △DEF by SAS
three parts used
then AC = DF by CPCTC
a part you never measured
You must establish congruence first, using three parts. Only then does CPCTC hand you the remaining three for free. Using it as a reason for the congruence itself is circular.
Figures often give the third part for free. These are the three free parts to look for.
△ABD and △CBD share side BD. AB = CB and AD = CD. Which test?
AB = CB, AD = CD
two sides given
BD = BD
a shared side equals itself
△ABD ≅ △CBD by SSS
three sides
A side both triangles use counts as a pair of equal sides (the reflexive property).
Segments AC and BD cross at E, with AE = CE and BE = DE. Which test?
AE = CE, BE = DE
two sides given
∠AEB = ∠CED
vertical angles, BETWEEN those sides
△AEB ≅ △CED by SAS
the angle is included
Crossing segments always hand you a pair of vertical angles; check whether it sits between the given sides.
In quadrilateral ABCD, AB ∥ DC and AB = DC. Diagonal AC is drawn. Which test?
AB = CD
given side
∠BAC = ∠DCA
alternate interior angles, AB ∥ DC
AC = CA
shared side
△BAC ≅ △DCA by SAS
the angle sits between AB and AC
Parallel lines supply equal angles. Write the congruence with matching letters in matching places: B↔D, A↔C.
Expect a figure or a description of three matching parts with the question "which test?", a "which is not a test" item (AAA, SSA), and a corresponding part read from a congruence statement. The fastest route is to list the three pairs you have — counting a shared side or vertical angles — then check whether the angle or side is included. The 10-second check: read the congruence letters in order and make sure each matched part sits in the same position in both names.
Which is not a test?
Which of these does not prove two triangles congruent?
AAA fixes only the shape, not the size — an equilateral triangle of side 2 and one of side 200 have identical angles. It proves similarity. The other three are valid congruence tests, and each of them involves at least one side, which is what pins the size down.
Hint: ask which option never mentions a side. Without a length, nothing fixes how big the triangle is.
