When a rigid motion carries $\triangle ABC$ onto $\triangle DEF$ (see Congruence and Rigid Motions), it sends each vertex to a specific vertex. The congruence statement $\triangle ABC \cong \triangle DEF$ records exactly that: $A \to D$, $B \to E$, $C \to F$.
Por qué funciona
The order of the letters is the correspondence. Matching letters in the same position gives the six pairs of corresponding parts, and each pair is equal because the rigid motion keeps lengths and angles. The picture can be turned or flipped any way; the letters still tell the truth.
| Angles | Sides |
|---|---|
| $\angle A \cong \angle D$ | $\overline{AB} \cong \overline{DE}$ |
| $\angle B \cong \angle E$ | $\overline{BC} \cong \overline{EF}$ |
| $\angle C \cong \angle F$ | $\overline{CA} \cong \overline{FD}$ |
Idea clave
In $\triangle ABC \cong \triangle DEF$, the first letters match, the second letters match and the third letters match. Read corresponding parts from the statement, not from where they sit on the page.
Ejemplo resuelto: Listing the parts
$\triangle PQR \cong \triangle XZY$. Which side corresponds to $\overline{QR}$, and which angle is congruent to $\angle R$?
- Pair the letters by position: $P \leftrightarrow X$, $Q \leftrightarrow Z$, $R \leftrightarrow Y$.
- $\overline{QR}$ matches $\overline{ZY}$.
- $\angle R$ matches $\angle Y$.
Respuesta: $\overline{QR} \cong \overline{ZY}$ and $\angle R \cong \angle Y$.
Ejemplo resuelto: Using the parts
$\triangle KLM \cong \triangle RST$, $m\angle K = 48^\circ$, $m\angle L = 75^\circ$ and $ST = 3x - 4$ while $LM = 11$. Find $m\angle T$ and $x$.
- $\angle T$ matches $\angle M$. In $\triangle KLM$, $m\angle M = 180 - 48 - 75 = 57^\circ$, so $m\angle T = 57^\circ$.
- $\overline{ST}$ matches $\overline{LM}$, so $3x - 4 = 11$, giving $3x = 15$ and $x = 5$.
Respuesta: $m\angle T = 57^\circ$ and $x = 5$.
When you write a congruence statement yourself, list the first triangle's vertices in any order, then list the second triangle's vertices in the matching order. $\triangle ABC \cong \triangle DEF$ and $\triangle BCA \cong \triangle EFD$ say the same thing; $\triangle ABC \cong \triangle EDF$ says something different.
Error común
Matching parts by where they sit on the page. If the second triangle is turned or flipped, its "bottom side" may not match the first triangle's bottom side. Always pair letters by their position in the congruence statement. Check: a statement like $\triangle ABC \cong \triangle BCA$ written for two triangles in the same spot is a warning sign; rewrite it carefully.