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Computable Euclid

Proposition 4

Theorem

Alternate name(s): SAS theorem.

If two triangles (△ABC, △DEF) have two sides (AB, AC) of one equal respectively to two sides (DE, DF) of the other, and also have the angles (∠BAC, ∠EDF) included by those sides equal, the triangles shall be congruent.

Commentary

1. Let △ABC and △DEF be two given triangles, with sides ￜABￜ = ￜDEￜ, ￜACￜ = ￜDFￜ. Let the angles ∠BAC and ∠EDF included by these two pairs of sides be equal.
2. These two triangles are said to be congruent, with the remaining sides and angles being equal, respectively.
3. Euclid didn't use the term "congruent". He expressed the same idea by saying that the triangles were "equal" (meaning equal areas) and that corresponding sides and angles were equal to each other.
4. This proposition is known as the SAS (side-angle-side) rule for triangle congruence.

Original statement

ἐὰν δύο τρίγωνα τὰς δύο πλϵυρὰς ταῖς δυσὶ πλϵυραῖς ἴσας ἔχῃ ἑκατέραν ἑκατέρᾳ καὶ τὴν γωνίαν τῇ γωνίᾳ ἴσην ἔχῃ τὴν ὑπὸ τῶν ἴσων ϵὐθϵιῶν πϵριϵχομένην, καὶ τὴν βάσιν τῇ βάσϵι ἴσην ἕξϵι, καὶ τὸ τρίγωνον τῷ τριγώνῳ ἴσον ἔσται, καὶ αἱ λοιπαὶ γωνίαι ταῖς λοιπαῖς γωνίαις ἴσαι ἔσονται ἑκατέρα ἑκατέρᾳ, ὑϕ᾽ ἃς αἱ ἴσαι πλϵυραὶ ὑποτϵίνουσιν.

English translation

If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend.

Computable version

Additional instances

Dependency graphs