Translations, reflections and rotations are called rigid motions. They give a precise meaning to “same size and same shape.”
Why it works
A rigid motion moves a figure without stretching or bending it, so it never changes a length or an angle. If a sequence of rigid motions carries one figure exactly onto another, then every side and every angle of one matches a side and an angle of the other. That is what we mean by congruent.
Key idea
Two figures are congruent if a sequence of rigid motions maps one onto the other. Corresponding sides and angles are equal. In $\triangle ABC \cong \triangle DEF$, the letters match in order: $A$ with $D$, $B$ with $E$, $C$ with $F$.
Worked examples
Worked example: Two moves in a row
Reflect $(2, 3)$ across the $y$-axis, then translate the image 4 units down.
- Reflect: $(2, 3) \to (-2, 3)$.
- Translate: $(-2, 3) \to (-2, -1)$.
Answer: $(-2, -1)$
Worked example: Show two triangles are congruent
$\triangle ABC$ has $A(1, 1)$, $B(4, 1)$, $C(1, 3)$; $\triangle DEF$ has $D(-1, -1)$, $E(-4, -1)$, $F(-1, -3)$. Are they congruent?
- Try a $180^\circ$ rotation about the origin, $(x, y) \to (-x, -y)$.
- It sends $A \to D$, $B \to E$ and $C \to F$.
- A rigid motion maps one onto the other.
Answer: Yes, $\triangle ABC \cong \triangle DEF$
Worked example: Use the matching parts
$\triangle PQR \cong \triangle XYZ$, with $PQ = 7$ and $\angle R = 40^\circ$. Find $XY$ and $\angle Z$.
- $PQ$ matches $XY$, so $XY = 7$.
- $\angle R$ matches $\angle Z$, so $\angle Z = 40^\circ$.
Answer: $XY = 7$, $\angle Z = 40^\circ$
Common mistake
Assuming the order of moves doesn’t matter. Reflecting across the $y$-axis and then translating 3 right sends $(1, 0)$ to $(2, 0)$, but translating first and then reflecting sends it to $(-4, 0)$. Do the moves in the order given.