Do you need all six pairs of corresponding parts from Corresponding Parts to know two triangles are congruent? No. Three well-chosen pairs force the rest. The trick is choosing the right three, and the proof uses the rigid motions you already know. You will use these criteria in proofs, so the writing skills from Writing a Proof matter here.
| Criterion | What must match | Why it works (sketch) |
|---|---|---|
| SSS | all three pairs of sides | the third vertex is where two circles cross; the motion has nowhere else to put it |
| SAS | two sides and the angle between them | the angle fixes the direction of the second side, and its length fixes where it ends |
| ASA | two angles and the side between them | the two angles fix two rays from the ends of the side, and two rays meet in one point |
Por qué funciona
Here is why SAS works. Suppose $AB = DE$, $AC = DF$ and $\angle A \cong \angle D$. Translate so $A$ lands on $D$. Rotate about $D$ so ray $AB$ lies along ray $DE$; since $AB = DE$, $B$ lands on $E$. If $C$ ends up on the other side of $\overline{DE}$ from $F$, reflect over $\overleftrightarrow{DE}$. Now the angles at $D$ match, so ray $AC$ lies along ray $DF$, and $AC = DF$ puts $C$ on $F$. All three vertices match: the triangles are congruent.
Ejemplo resuelto: Choosing the criterion
In $\triangle ABC$ and $\triangle DEF$, $\overline{AB} \cong \overline{DE}$, $\angle B \cong \angle E$ and $\overline{BC} \cong \overline{EF}$. Are they congruent? By what?
- The marked angle, $\angle B$, is where the marked sides $\overline{AB}$ and $\overline{BC}$ meet: it is the included angle.
- Two sides and the included angle: SAS.
Respuesta: Yes, $\triangle ABC \cong \triangle DEF$ by SAS.
Ejemplo resuelto: What is missing?
$\angle A \cong \angle D$ and $\angle C \cong \angle F$. Which pair of sides would prove the triangles congruent by ASA?
- ASA needs the side between the two angles.
- $\overline{AC}$ joins $A$ and $C$; $\overline{DF}$ joins $D$ and $F$.
Respuesta: $\overline{AC} \cong \overline{DF}$.
Idea clave
SSS, SAS and ASA each prove two triangles congruent. In SAS the angle must be between the two sides; in ASA the side must be between the two angles.
Error común
Accepting SSA or AAA. With SSA (an angle that is not between the two sides), two different triangles can fit, as the picture above shows. With AAA, a triangle and an enlarged copy have the same angles but different sides. Check where the angle sits relative to the sides before naming a criterion.