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Euclid · Elements §1.prop.24-1.prop.25

Relationship between Included Angles and Bases in Two Triangles

Passage 14 of 316 · Greek

Summary

These propositions prove that if two triangles have two sides equal to two sides respectively, the one with the greater included angle has the greater base (Proposition 24), and conversely, the one with the greater base has the greater included angle (Proposition 25).

§1.prop.24ἐὰν δύο τρίγωνα τὰς δύο πλευρὰς δύο πλευραῖς ἴσας ἔχῃ ἑκατέραν ἑκατέρᾳ, τὴν δὲ γωνίαν τῆς γωνίας μείζονα ἔχῃ τὴν ὑπὸ τῶν ἴσων εὐθειῶν περιεχομένην, καὶ τὴν βάσιν τῆς βάσεως μείζονα ἕξει.
If two triangles have two sides equal to two sides respectively, and have the angle contained by the equal straight lines greater than the angle, they will also have the base greater than the base.
ἔστω δύο τρίγωνα τὰ ΑΒΓ, ΔΕΖ τὰς δύο πλευρὰς τὰς ΑΒ, ΑΓ ταῖς δύο πλευραῖς ταῖς ΔΕ, ΔΖ ἴσας ἔχοντα ἑκατέραν ἑκατέρᾳ, τὴν μὲν ΑΒ τῇ ΔΕ τὴν δὲ ΑΓ τῇ ΔΖ, ἡ δὲ πρὸς τῷ Α γωνία τῆς πρὸς τῷ Δ γωνίας μείζων ἔστω· λέγω, ὅτι καὶ βάσις ἡ ΒΓ βάσεως τῆς ΕΖ μείζων ἐστίν.
Let there be two triangles ABC, DEF having two sides AB, AC equal to two sides DE, DF respectively, namely AB to DE, and AC to DF, and let the angle at A be greater than the angle at D; I say that the base BC is also greater than the base EF.
ἐπεὶ γὰρ μείζων ἡ ὑπὸ ΒΑΓ γωνία τῆς ὑπὸ ΕΔΖ γωνίας, συνεστάτω πρὸς τῇ ΔΕ εὐθείᾳ καὶ τῷ πρὸς αὐτῇ σημείῳ τῷ Δ τῇ ὑπὸ ΒΑΓ γωνίᾳ ἴση ἡ ὑπὸ ΕΔΗ, καὶ κείσθω ὁποτέρᾳ τῶν ΑΓ, ΔΖ ἴση ἡ ΔΗ, καὶ ἐπεζεύχθωσαν αἱ ΕΗ, ΖΗ. ἐπεὶ οὖν ἴση ἐστὶν ἡ μὲν ΑΒ τῇ ΔΕ, ἡ δὲ ΑΓ τῇ ΔΗ, δύο δὴ αἱ ΒΑ, ΑΓ δυσὶ ταῖς ΕΔ, ΔΗ ἴσαι εἰσὶν ἑκατέρα ἑκατέρᾳ·
For, since the angle BAC is greater than the angle EDF, let there be constructed on the straight line DE, and at the point D on it, the angle EDH equal to the angle BAC; and let DH be made equal to either of the AC, DF, and let EH, FH be joined.
καὶ γωνία ἡ ὑπὸ ΒΑΓ γωνίᾳ τῇ ὑπὸ ΕΔΗ ἴση· βάσις ἄρα ἡ ΒΓ βάσει τῇ ΕΗ ἐστιν ἴση.
Since therefore AB is equal to DE, and AC to DH, the two BA, AC are equal to the two ED, DH, each to each; and the angle BAC is equal to the angle EDH; therefore the base BC is equal to the base EH.
πάλιν, ἐπεὶ ἴση ἐστὶν ἡ ΔΖ τῇ ΔΗ, ἴση ἐστὶ καὶ ἡ ὑπὸ ΔΗΖ γωνία τῇ ὑπὸ ΔΖΗ· μείζων ἄρα ἡ ὑπὸ ΔΖΗ τῆς ὑπὸ ΕΗΖ· πολλῷ ἄρα μείζων ἐστὶν ἡ ὑπὸ ΕΖΗ τῆς ὑπὸ ΕΗΖ. καὶ ἐπεὶ τρίγωνόν ἐστι τὸ ΕΖΗ μείζονα ἔχον τὴν ὑπὸ ΕΖΗ γωνίαν τῆς ὑπὸ ΕΗΖ, ὑπὸ δὲ τὴν μείζονα γωνίαν ἡ μείζων πλευρὰ ὑποτείνει, μείζων ἄρα καὶ πλευρὰ ἡ ΕΗ τῆς ΕΖ. ἴση δὲ ἡ ΕΗ τῇ ΒΓ· μείζων ἄρα καὶ ἡ ΒΓ τῆς ΕΖ. ἐὰν ἄρα δύο τρίγωνα τὰς δύο πλευρὰς δυσὶ πλευραῖς ἴσας ἔχῃ ἑκατέραν ἑκατέρᾳ, τὴν δὲ γωνίαν τῆς γωνίας μείζονα ἔχῃ τὴν ὑπὸ τῶν ἴσων εὐθειῶν περιεχομένην, καὶ τὴν βάσιν τῆς βάσεως μείζονα ἕξει· ὅπερ ἔδει δεῖξαι.
Again, since DF is equal to DH, the angle DHZ is also equal to the angle DZH; therefore the angle DZH is greater than the angle EHZ; therefore the angle EZH is much greater than the angle EHZ. And since EZH is a triangle having the angle EZH greater than the angle EHZ, and the greater side subtends the greater angle, therefore the side EH is also greater than EZ. But EH is equal to BC; therefore BC is also greater than EZ. If therefore two triangles have two sides equal to two sides respectively, and have the angle contained by the equal straight lines greater than the angle, they will also have the base greater than the base; which was to be proved.
§1.prop.25ἐὰν 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.
If two triangles have two sides equal to two sides respectively, and have the base greater than the base, they will also have the angle contained by the equal straight lines greater than the angle.
ἔστω δύο τρίγωνα τὰ ΑΒΓ, ΔΕΖ τὰς δύο πλευρὰς τὰς ΑΒ, ΑΓ ταῖς δύο πλευραῖς ταῖς ΔΕ, ΔΖ ἴσας ἔχοντα ἑκατέραν ἑκατέρᾳ, τὴν μὲν ΑΒ τῇ ΔΕ, τὴν δὲ ΑΓ τῇ ΔΖ· βάσις δὲ ἡ ΒΓ βάσεως τῆς ΕΖ μείζων ἔστω· λέγω, ὅτι καὶ γωνία ἡ ὑπὸ ΒΑΓ γωνίας τῆς ὑπὸ ΕΔΖ μείζων ἐστίν·
Let there be two triangles ABC, DEF having two sides AB, AC equal to two sides DE, DF respectively, namely AB to DE, and AC to DF; and let the base BC be greater than the base EF; I say that the angle BAC is also greater than the angle EDF.
εἰ γὰρ μή, ἤτοι ἴση ἐστὶν αὐτῇ ἢ ἐλάσσων·
For, if not, it is either equal to it or less.
ἴση μὲν οὖν οὐκ ἔστιν ἡ ὑπὸ ΒΑΓ τῇ ὑπὸ ΕΔΖ· ἴση γὰρ ἂν ἦν καὶ βάσις ἡ ΒΓ βάσει τῇ ΕΖ· οὐκ ἔστι δέ.
Now the angle BAC is not equal to the angle EDF; for then the base BC would also have been equal to the base EF; but it is not.
οὐκ ἄρα ἴση ἐστὶ γωνία ἡ ὑπὸ ΒΑΓ τῇ ὑπὸ ΕΔΖ·
Therefore the angle BAC is not equal to the angle EDF.
οὐδὲ μὴν ἐλάσσων ἐστὶν ἡ ὑπὸ ΒΑΓ τῆς ὑπὸ ΕΔΖ· ἐλάσσων γὰρ ἂν ἦν καὶ βάσις ἡ ΒΓ βάσεως τῆς ΕΖ· οὐκ ἔστι δέ·
Neither indeed is the angle BAC less than the angle EDF; for then the base BC would also have been less than the base EF; but it is not.
οὐκ ἄρα ἐλάσσων ἐστὶν ἡ ὑπὸ ΒΑΓ γωνία τῆς ὑπὸ ΕΔΖ. ἐδείχθη δὲ ὅτι οὐδὲ ἴση· μείζων ἄρα ἐστὶν ἡ ὑπὸ ΒΑΓ τῆς ὑπὸ ΕΔΖ. ἐὰν ἄρα δύο τρίγωνα τὰς δύο πλευρὰς δυσὶ πλευραῖς ἴσας ἔχῃ ἑκατέραν ἑκάτερᾳ, τὴν δὲ βάσιν τῆς βάσεως μείζονα ἔχῃ, καὶ τὴν γωνίαν τῆς γωνίας μείζονα ἕξει τὴν ὑπὸ τῶν ἴσων εὐθειῶν περιεχομένην· ὅπερ ἔδει δεῖξαι.
Therefore the angle BAC is not less than the angle EDF. And it was proved that it is not equal either; therefore the angle BAC is greater than the angle EDF. If therefore two triangles have two sides equal to two sides respectively, and have the base greater than the base, they will also have the angle contained by the equal straight lines greater than the angle; which was to be proved.

Notes

  1. 1.prop.24τὴν ὑπὸ τῶν ἴσων εὐθειῶν περιεχομένην — The present passive participle περιεχομένην feminine accusative singular modifies the preceding τὴν γωνίαν, accompanied by the prepositional phrase ὑπό with the genitive (τῶν ἴσων εὐθειῶν) to denote the agent: 'the angle contained by the equal straight lines'.
  2. 1.prop.24συνεστάτω πρὸς τῇ ΔΕ εὐθείᾳ καὶ τῷ πρὸς αὐτῇ σημείῳ τῷ Δ — The third-person singular perfect imperative of συνίστημι, expressing a demand for the completed state of the construction. The preposition πρός with the dative indicates the location/basis where the construction is to occur ('on the straight line... and at the point...').
  3. 1.prop.24ὁποτέρᾳ τῶν ΑΓ, ΔΖ ἴση — The feminine dative singular ὁποτέρᾳ ('either of the two') is governed by the adjective ἴση ('equal to'). The genitive plural τῶν ΑΓ, ΔΖ serves as a partitive genitive.
  4. 1.prop.24πολλῷ ἄρα μείζων ἐστὶν ἡ ὑπὸ ΕΖΗ τῆς ὑπὸ ΕΗΖ — The dative πολλῷ functions as a dative of degree of difference with the comparative μείζων, meaning 'therefore the angle EZH is much greater than the angle EHZ'.
  5. 1.prop.25εἰ γὰρ μή, ἤτοι ἴση ἐστὶν αὐτῇ ἢ ἐλάσσων — The phrase εἰ γὰρ μή is an elliptical conditional clause ('for if not' / 'if the angle BAC is not greater than the angle EDF'). The correlative conjunctions ἤτοι ... ἤ ... introduce an exclusive disjunction ('either equal to it or less').

Cite this passage

Euclid, Elements §1.prop.24-1.prop.25. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:1.prop.24-1.prop.25

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