IV. SAS similarity axiom

SAS stands for Side-Angle-Side, meaning "two sides and their included angle".

Statement

Let $\triangle ABC$ and $\triangle A'B'C'$ be two triangles. If

$$\frac{|A'B'|}{|AB|} \;=\; \frac{|A'C'|}{|AC|} \;=\; k$$

and the included angles satisfy

$$\angle BAC \;=\; \angle B'A'C',$$

then

$$\triangle ABC \;\sim\; \triangle A'B'C'.$$

Unpacking "$\sim$" (similarity) gives two things: corresponding sides are in proportion

$$\frac{|A'B'|}{|AB|} \;=\; \frac{|A'C'|}{|AC|} \;=\; \frac{|B'C'|}{|BC|} \;=\; k,$$

and corresponding angles are equal

$$\angle A = \angle A',\quad \angle B = \angle B',\quad \angle C = \angle C'.$$

The axiom only uses two of those proportions plus one included angle; the remaining proportion and the other two angles are what the axiom asserts.

Intuition

The axiom makes a very strong claim: once you nail down the ratios of two sides and the angle between them, the shape of the triangle is nailed down.

Think of $\triangle ABC$ as a rubber triangle: grab vertex $A$, stretch both sides $AB$ and $AC$ by a factor of $k$ at the same time, and require that the apex angle $\angle BAC$ stay the same. Then $B$ and $C$ land at new positions $B'$ and $C'$, and the opposite side $B'C'$ — which you did nothing to set — automatically has length $k\cdot|BC|$, with the right direction, and the other two angles are copied along too.

Put differently, there are nominally six free parameters (three sides plus three angles), but "two-side ratio plus included angle" only locks three numbers (in fact only two independent ones, since both ratios equal $k$, plus one angle); the remaining three numbers are forced by the axiom. That is why this axiom is so "expensive" — in a single sentence it buys away three degrees of freedom.

Why this is an axiom, not a theorem

Among the four axioms this is the deepest. It is not describing "how to measure" — it is building a bridge: welding together the already-existing "world of lengths" and "world of angles". The four points below explain why this work has to be done as an axiom.

1. Euclid cannot prove SAS — he only "moves it on top"

Proposition 4 of Book I of the Elements tries to prove SAS (side-angle-side) congruence: pick up the whole $\triangle ABC$ and slide it over so that $A$ lands on $A'$ and $AB$ lies along $A'B'$; since $|AB| = |A'B'|$, $B$ must land on $B'$; and since $\angle A = \angle A'$, $AC$ must lie along $A'C'$, so $C$ lands on $C'$; finally, by "two points determine a line", $BC$ coincides with $B'C'$.

This argument has an unstated premise: rigid motions exist, and they preserve length and angle. Euclid had no concept of rigid motion — he is in fact tacitly using a kind of "geometric intuition", substituting the act of moving figures for a proof. Later mathematicians (such as Bertrand Russell) criticized this step as one of the Elements' biggest logical gaps.

2. Hilbert's price: seven extra congruence axioms

Hilbert (1899), to make "superposition" rigorous, wrote a dedicated set of congruence axioms (Group III, with 5 segment-congruence axioms + 1 angle-congruence axiom + 1 SAS congruence axiom). This group of axioms posits SAS congruence directly as an axiom, no longer trying to "prove" it — it merely decomposes the intuition of "rigid motion" into several finer algebraic properties.

The price: more axioms, in finer pieces. But the spirit is honest — SAS cannot be derived from anything simpler.

3. Birkhoff's choice: adopt it directly, and upgrade to "similarity"

Birkhoff (1932) accepted the fact that "SAS cannot be proved" and simply listed it as an axiom, while also strengthening the proposition: not only "equality" holds, proportionality does too. This has two benefits:

• Congruence becomes a special case. Take $k = 1$, and similarity degenerates to congruence. In the Elements and Hilbert one has to discuss SAS congruence and the similarity theorems separately; Birkhoff handles both in one stroke.
• Direct interface with the real-number scale. Axiom I has already loaded length into the real numbers, and the ratio $k$ is a natural product of real-number division. Upgrading "equal length" ($k = 1$) to "equal ratio" ($k \in \mathbb{R}_{>0}$) adds no cognitive overhead.

4. It welds the "world of lengths" and the "world of angles" together

Looking back at the structure of the four axioms:

• I · Ruler: line $\leftrightarrow \mathbb{R}$, providing length.
• III · Protractor: ray $\leftrightarrow [0, 2\pi)$, providing angle.

With I and III alone, length and angle are two systems that don't talk to each other — measuring $|AB| = 3$ and $\angle BAC = 40^\circ$ creates no forced relationship between those two numbers.

The SAS similarity axiom is precisely the bridge that pins the two systems together. It says: in two triangles, if the length ratios of two sides match and the angles between them match, then —

• the length ratio of the third side automatically matches (length obeys angle);
• the other two angles automatically match (angle obeys length).

Length differences and angle differences are now mutually constrained: you cannot freely change a length ratio without also moving an angle, and vice versa. This is the source of Euclidean geometry's "rigidity".

In practice: WLOG "we may assume the two triangles share a vertex"

WLOG = Without Loss Of Generality. It means: in a proof we make some simplifying assumption (e.g. "translate and rotate the two triangles so they share a vertex"), and that simplification does not cost the conclusion any generality — because the general case, stripped of the simplification, can always be reduced back to the simplified case via the same transformation. The legitimacy of the move hinges on one thing: the transformation preserves the quantities you care about (here: side ratios + included angle).

Later proofs repeatedly say "we may translate and rotate $\triangle A'B'C'$ so that $A'$ coincides with $A$ and $\angle B'A'C'$ has the same orientation as $\angle BAC$" — this uses the isometry group (translations + rotations + reflections) already given to us by axioms I + III to drag the two triangles into a shared-vertex position, where the SAS similarity axiom can act directly. Concretely:

• two arbitrary triangles initially sit anywhere in the plane;
• an isometry drags $A'$ to $A$ and rotates $\angle B'A'C'$ to align with $\angle BAC$;
• this "repositioning" preserves side lengths and included angles, so proving the shared-vertex case ⇔ proving the general case.

The legitimacy of this "WLOG" maneuver corresponds exactly to the two hypotheses of the SAS axiom (side ratios + included angle).

In one sentence: the length system and the angle system are otherwise unrelated; the SAS similarity axiom is the weld that fuses them. Once welded, all of Euclidean geometry can stand up.

Immediate consequences

Several facts read straight off the axiom, no further proof needed:

• Third-side ratio: $\dfrac{|B'C'|}{|BC|} = k$.
• The other two angles are equal: $\angle B = \angle B'$, $\angle C = \angle C'$.
• Perimeter ratio $= k$: $\dfrac{|A'B'|+|B'C'|+|C'A'|}{|AB|+|BC|+|CA|} = k$.
• Ratio of corresponding altitudes / medians / angle bisectors $= k$: every "linear length" scales by the same factor.
• The special case $k = 1$ is SAS congruence: two equal sides + equal included angle $\Rightarrow$ the two triangles are congruent.

These five facts are the foundation of every later similarity / congruence argument.

Compared with the Elements / Hilbert

Euclid's Elements (I.4) uses the "superposition method" to prove SAS congruence: move one triangle onto the other and check coincidence. This step silently uses "rigid motions exist and preserve geometry", but the Elements has no concept of rigid motion. As a result, this proposition has long been regarded as one of Euclid's least rigorous proofs. Euclid also has no "similarity" as an independent axiom — he detours through "the theory of proportion" in Books V and VI before he can talk about similar triangles.

Hilbert (1899) uses Group III congruence axioms to break "superposition" down into several purely algebraic properties (transitivity of segment congruence, transitivity of angle congruence, SAS congruence, etc.), at the price of an inflated axiom count (the congruence group alone has 5–7 axioms). In Hilbert's system SAS congruence is an axiom, and similarity requires developing a separate theory of proportion before it can be discussed.

Birkhoff (1932) simply upgrades the proposition to SAS similarity and takes it as a single axiom. The price is treating real numbers as already constructed (axiom I supplies them); the payoff:

• one axiom handles both congruence and similarity;
• no reliance on any "superposition" or "rigid motion" geometric intuition;
• honesty toward the student: this fact is simply adopted, not derived.

All three systems acknowledge that SAS cannot be proved; they differ only in how many axioms they spend explaining it, and at which level (congruence / similarity) they state it.

What it unlocks

This axiom is the engine of middle-school geometry. The following theorems all depend on it, directly or indirectly:

• SSS / ASA / AAS / HL (hypotenuse-leg) congruence criteria: take $k = 1$ and derive them from SAS similarity + axioms I, III.
• AA similarity: two pairs of equal angles $\Rightarrow$ the two triangles are similar. The proof uses an auxiliary construction + SAS.
• SSS similarity criterion: three sides in proportion $\Rightarrow$ similar.
• Basic proportionality (intercept theorem): a line parallel to one side of a triangle cuts the other two sides into proportional segments.
• Triangle midsegment theorem: the midsegment is parallel to the third side and half its length.
• Pythagorean theorem: the cleanest proof goes via "the altitude to the hypotenuse + three pairs of similar triangles" — a direct product of SAS similarity.
• Power-of-a-point theorems (secants / tangents / intersecting chords): at the core, corresponding sides of similar triangles are in proportion.
• Area ratio of similar triangles $= k^2$: sides scale by $k$, so areas scale by $k^2$.
• Inscribed angle theorem / Tangent–chord angle = inscribed / criterion for four concyclic points: every "angle equals angle" theorem ultimately relies on similar triangles to translate "equal angles" into "equal ratios" and back.

Among the four axioms: I provides length, III provides angle, II gives "two points determine a line", and IV connects the previous three so that "length" and "angle" mutually constrain each other. Once this axiom is adopted, every standard theorem of middle-school geometry sits downstream of it.