Congruent figures
Two figures are congruent if one can be placed exactly on top of the other, so they have the same shape AND the same size. All corresponding sides are equal and all corresponding angles are equal. We write $\triangle ABC\cong\triangle PQR$.
Conditions for congruent triangles
We do not need all six measurements to be sure two triangles are congruent. Any one of these tests is enough:
- SSS — three sides equal;
- SAS — two sides and the included angle equal;
- ASA — two angles and the included side equal;
- RHS — right angle, hypotenuse and one side equal (right-angled triangles).
Remember
AAA is NOT a congruence test. Equal angles only guarantee the same shape (similarity), not the same size.
Key formula
$\angle A+\angle B+\angle C=180^\circ\qquad\text{congruent}\iff\text{similar with scale factor }k=1$
Congruency versus similarity
Similar figures have the same shape but not necessarily the same size — corresponding angles are equal and corresponding sides are in a fixed ratio (the scale factor). Congruency is the special case of similarity where the scale factor is $1$.
Example
If $\triangle ABC\cong\triangle PQR$ with $\angle A=40^\circ$ and $\angle B=75^\circ$, then $\angle C=180^\circ-40^\circ-75^\circ=65^\circ$, so the corresponding angle $\angle R=65^\circ$ and side $PQ=AB$.
Remember
Congruent $\Rightarrow$ similar with scale factor $1$; similar does not imply congruent.