Congruence and Similarity

Two triangles are congruent by:

• SSS — three sides equal;
• SAS — two sides and the angle between them;
• ASA / AAS — two angles and any matching side;
• RHS — right angle, hypotenuse and one other side.

AAA gives only similarity. SSA (the angle not between the sides) is not a rule, except for the right-angled case RHS.

• AA: two pairs of equal angles make triangles similar.
• If the scale factor is $k$, every corresponding length (side, perimeter, altitude, median, bisector) is in the ratio $k$.
• Shared angle: if $D$ is on $BC$ and $\angle ADC = \angle BAC$, then triangles $ADC$ and $BAC$ share $\angle C$, so they are similar and $\dfrac{DC}{AC} = \dfrac{AC}{BC}$, i.e. $AC^2 = DC\cdot BC$.
• In a statement like $\triangle ABR \sim \triangle PQR$, the vertices match in the order written: $A$ with $P$, $B$ with $Q$, $R$ with $R$.

Poles of heights $a$ and $b$ stand a distance $d$ apart; the crossing point is at height $h$ and at distance $x$ from the foot of the first pole.

• Similar triangles with the second pole: $\dfrac hb = \dfrac xd$.
• Similar triangles with the first pole: $\dfrac ha = \dfrac{d - x}{d}$.
• Adding: $\dfrac ha + \dfrac hb = 1$, so $h = \dfrac{ab}{a + b}$, and $d$ has cancelled.