相似と合同

Congruent figures have all corresponding sides equal and all corresponding angles equal. We write this with the symbol $\cong$, as in $△ABC\cong △DEF$. When stating a congruence it is important to list corresponding vertices in the same order ($A\leftrightarrow D$, $B\leftrightarrow E$, $C\leftrightarrow F$). For triangles we do not need to check all six elements (three sides and three angles): certain sets of three matching pieces are enough to guarantee congruence.

There are four conditions that prove two triangles congruent: SSS (all three sides equal), SAS (two sides and the included angle equal), ASA (two angles and the included side equal — and since fixing two angles fixes the third, AAS works the same way), and RHS (in right-angled triangles, the hypotenuse and one other side equal). Note that SSA (two sides and a non-included angle) and AAA (three angles only) are not congruence conditions. AAA gives the same shape but possibly a different size, so it is a condition for similarity instead.

In similar figures, all corresponding angles are equal and the ratios of all corresponding sides are equal. This common ratio is the scale factor $k$. For example, if $△ABC\sim △DEF$ and each corresponding side is twice as long, then $k=2$. Triangles can be shown similar by AA (two pairs of equal angles), by SSS similarity (all three side ratios equal), or by SAS similarity (two side ratios equal with the included angle equal). We use the symbol $\sim$.

Once two triangles are known to be similar, we use the equality of corresponding side ratios to find an unknown side. The steps are: (1) match corresponding vertices and sides, (2) write a proportion equating corresponding side ratios, and (3) solve it. For instance, from $\frac{x}{6}=\frac{15}{10}$ we multiply both sides by $6$ to get $x=6\times \frac{15}{10}=9$. The key is to pair the correct sides — use the equal angles to match them, rather than being misled by how the diagram is drawn.

For similar figures with a length ratio (scale factor) of $k$, the ratio of areas is $k^2$ and the ratio of volumes is $k^3$. So if lengths double, areas multiply by $2^2=4$ and volumes by $2^3=8$. Conversely, if the area ratio is $9:16$, the length ratio is the square root, $3:4$, and the volume ratio is the cube of that, $27:64$. Keep firmly in mind: length to area means squaring, and length to volume means cubing.