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Computable Euclid

Proposition 25

Theorem

If two triangles (△ABC, △DEF) have two sides (AB, AC) of one respectively equal to two sides (DE, DF) of the other, but have the base of one longer than the base of the other (ￜBCￜ > ￜEFￜ), then the angle contained by the sides of that which has the longer base is bigger than the angle contained by the sides of the other (∠BAC > ∠EDF).

Commentary

  • Let △ABC and △DEF be two given triangles, and let two pairs of sides of these two triangles be equal, namely ￜABￜ = ￜDEￜ and ￜACￜ = ￜDFￜ.
  • Let BC (the base of △ABC) be longer than EF (the base of △DEF).
  • Then ∠BAC (contained by AB and AC and opposite to BC in △ABC) is bigger than ∠EDF (contained by DE and DF and opposite to EF in △DEF).
  • This proposition is the converse of the previous proposition, Book 1 Proposition 24.
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    Original statement

    ἐὰν δύο τρίγωνα τὰς δύο πλϵυρὰς δυσὶ πλϵυραῖς ἴσας ἔχῃ ἑκατέραν ἑκατέρᾳ, τὴν δὲ βάσιν τῆς βάσϵως μϵίζονα ἔχῃ, καὶ τὴν γωνίαν τῆς γωνίας μϵίζονα ἕξϵι τὴν ὑπὸ τῶν ἴσων ϵὐθϵιῶν πϵριϵχομένην.

    English translation

    If two triangles have the two sides equal to two sides respectively, but have the base greater than the base, they will also have the one of the angles contained by the equal straight lines greater than the other.

    Computable version

    Additional instances

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