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Computable Euclid

Proposition 26a

Theorem

Alternate name(s): ASA theorem.

If two triangles (△ABC, △DEF) have two angles of one equal respectively to two angles of the other (∠ABC = ∠DEF, ∠ACB = ∠DFE), and the sides between the corresponding angles are equal (ￜBCￜ = ￜEFￜ), then the triangles are congruent.

Commentary

1. Let △ABC and △DEF be two given triangles, with two pairs of corresponding angles being equal, namely ∠ABC = ∠DEF and ∠ACB = ∠DFE. Let the sides BC and EF between the corresponding angles be equal.
2. These two triangles are said to be congruent, with the remaining sides and angles being equal, respectively.
3. This proposition is known as the ASA (or side-angle-side) rule for triangle congruence.
4. Euclid's statement of proposition 26 specified two cases for the position of the pair of equal sides. This proposition is one case and Book 1 Proposition 26b is the other.

Original statement

ἐὰν δύο τρίγωνα τὰς δύο γωνίας δυσὶ γωνίαις ἴσας ἔχῃ ἑκατέραν ἑκατέρᾳ καὶ μίαν πλϵυρὰν μιᾷ πλϵυρᾷ ἴσην ἤτοι τὴν πρὸς ταῖς ἴσαις γωνίαις ἢ τὴν ὑποτϵίνουσαν ὑπὸ μίαν τῶν ἴσων γωνιῶν, καὶ τὰς λοιπὰς πλϵυρὰς ταῖς λοιπαῖς πλϵυραῖς ἴσας ἕξϵι ἑκατέραν ἑκατέρᾳ καὶ τὴν λοιπὴν γωνίαν τῇ λοιπῇ γωνίᾳ.

English translation

If two triangles have the two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that subtending one of the equal angles, they will also have the remaining sides equal to the remaining sides and the remaining angle to the remaining angle.

Computable version

Additional instances

Dependency graphs