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ἐὰν τριγώνου αἱ δύο γωνίαι ἴσαι ἀλλήλαις ὦσιν, καὶ αἱ ὑπὸ τὰς ἴσας γωνίαις ὑποτείνουσαι πλευραὶ ἴσαι ἀλλήλαις ἔσονται.

ἔστω τρίγωνον τὸ ΑΒΓ ἴσην ἔχον τὴν ὑπὸ ΑΒΓ γωνίαν
5τῇ ὑπὸ ΑΓΒ γωνίᾳ: λέγω, ὅτι καὶ πλευρὰ ΑΒ πλευρᾷ τῇ ΑΓ ἐστιν ἴση.

εἰ γὰρ ἄνισός ἐστιν ΑΒ τῇ ΑΓ, ἑτέρα αὐτῶν μείζων ἐστίν. ἔστω μείζων ΑΒ, καὶ ἀφῃρήσθω ἀπὸ τῆς μείζονος τῆς ΑΒ
10τῇ ἐλάττονι τῇ ΑΓ ἴση ΔΒ, καὶ ἐπεζεύχθω ΔΓ.

ἐπεὶ οὖν ἴση ἐστὶν ΔΒ τῇ ΑΓ κοινὴ δὲ ΒΓ, δύο δὴ αἱ ΔΒ, ΒΓ δύο ταῖς ΑΓ, ΓΒ ἴσαι εἰσὶν ἑκατέρα ἑκατέρᾳ, καὶ γωνία ὑπὸ ΔΒΓ
15γωνίᾳ τῇ ὑπὸ ΑΓΒ ἐστιν ἴση: βάσις ἄρα ΔΓ βάσει τῇ ΑΒ ἴση ἐστίν, καὶ τὸ ΔΒΓ τρίγωνον τῷ ΑΓΒ τριγώνῳ ἴσον ἔσται, τὸ ἔλασσον τῷ μείζονι: ὅπερ ἄτοπον: οὐκ ἄρα ἄνισός ἐστιν ΑΒ τῇ ΑΓ: ἴση ἄρα.

ἐὰν ἄρα τριγώνου αἱ δύο γωνίαι ἴσαι ἀλλήλαις ὦσιν,
20καὶ αἱ ὑπὸ τὰς ἴσας γωνίας ὑποτείνουσαι πλευραὶ ἴσαι ἀλλήλαις ἔσονται: ὅπερ ἔδει δεῖξαι.

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