Proposition 26b
Theorem
Alternate name(s): AAS 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 opposite one pair of equal angles are equal (ￜACￜ = ￜDFￜ ), 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 AC and DF opposite to one pair of equal angles (∠ABC and ∠DEF ) 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 AAS (or angle-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 26a is the other.
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 AAS (or angle-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 26a 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.
