SSS, SAS and ASA gave you SSS, SAS and ASA. Two more criteria save time: AAS for any triangle, and HL for right triangles.
Why it works
AAS: if two angles of one triangle match two angles of another, the third angles match too, because each set adds to $180^\circ$. So a side that is not between the two given angles is still between two matching angles, and ASA applies.
HL: in right triangles, if the hypotenuses match and one pair of legs matches, the Pythagorean theorem makes the other legs match: $\sqrt{c^2 - a^2}$ is the same for both. Then SSS applies.
HL: in right triangles, if the hypotenuses match and one pair of legs matches, the Pythagorean theorem makes the other legs match: $\sqrt{c^2 - a^2}$ is the same for both. Then SSS applies.
Key idea
AAS: two angles and a side that is not between them. HL: right triangles with congruent hypotenuses and one pair of congruent legs. Both prove congruence.
Worked example: AAS
$\angle A \cong \angle D$, $\angle B \cong \angle E$ and $\overline{BC} \cong \overline{EF}$. Which criterion applies?
- The side $\overline{BC}$ touches $\angle B$ but not $\angle A$: it is not between the two angles, so this is not ASA.
- Two angles and a non-included side: AAS.
Answer: $\triangle ABC \cong \triangle DEF$ by AAS.
Worked example: HL with numbers
Right triangles $PQR$ and $STU$ have right angles at $R$ and $U$. $PQ = ST = 13$ and $PR = SU = 5$. Show they are congruent and find $TU$.
- $\overline{PQ}$ and $\overline{ST}$ are opposite the right angles: they are the hypotenuses, and they are congruent.
- $\overline{PR}$ and $\overline{SU}$ are matching legs, also congruent. So $\triangle PQR \cong \triangle STU$ by HL.
- $TU = QR = \sqrt{13^2 - 5^2} = \sqrt{144} = 12$.
Answer: Congruent by HL, and $TU = 12$.
Worked example: Choosing among five
Two triangles have $\angle A \cong \angle D$, $\overline{AB} \cong \overline{DE}$, $\angle B \cong \angle E$. Is this AAS or ASA?
- The side $\overline{AB}$ joins the two marked angles, $A$ and $B$.
Answer: ASA (the side is between the angles).
Common mistake
Using HL for triangles that are not right triangles. Hypotenuse-leg is really SSA, which fails in general; it only works because the right angle makes the Pythagorean theorem available. Check for a right angle (given or marked) before writing HL.