PRINCIPIA · THEOREM

SSS / SAS similarity tests

Dependencies: SAS similarity axiom, SSS congruence.

Statement

Suppose △ABC\triangle ABC and △A′B′C′\triangle A'B'C' have vertices listed in corresponding order. Both of the following give a test for "the two triangles are similar":

(SSS similarity)\quadthree corresponding sides in proportion:

∣AB∣∣A′B′∣  =  ∣BC∣∣B′C′∣  =  ∣CA∣∣C′A′∣  =  k⟹△ABC∼△A′B′C′.\dfrac{|AB|}{|A'B'|} \;=\; \dfrac{|BC|}{|B'C'|} \;=\; \dfrac{|CA|}{|C'A'|} \;=\; k \quad\Longrightarrow\quad \triangle ABC \sim \triangle A'B'C'.

(SAS similarity)\quadone pair of corresponding angles equal + the two sides forming that angle in proportion:

∠A=∠A′,∣AB∣∣A′B′∣  =  ∣CA∣∣C′A′∣  =  k⟹△ABC∼△A′B′C′.\angle A = \angle A',\quad \dfrac{|AB|}{|A'B'|} \;=\; \dfrac{|CA|}{|C'A'|} \;=\; k \quad\Longrightarrow\quad \triangle ABC \sim \triangle A'B'C'.

SAS similarity is precisely the form of axiom IV itself; the entire proof below focuses on SSS similarity.

A summary diagram of SSS similarity: three pairs of side ratios equal ⇒ the two triangles are similar.

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