Euclid · Elements §1.prop.14-1.prop.15

Conditions for a Straight Line and Vertical Angles

Passage 10 of 316 · Greek

Summary

In Proposition 14, it is proved that if two adjacent angles equal two right angles, their outer sides form a straight line, and in Proposition 15, it is shown that when two straight lines intersect, the vertical angles are equal to each other.

§1.prop.14ἐὰν πρός τινι εὐθείᾳ καὶ τῷ πρὸς αὐτῇ σημείῳ δύο εὐθεῖαι μὴ ἐπὶ τὰ αὐτὰ μέρη κείμεναι τὰς ἐφεξῆς γωνίας δυσὶν ὀρθαῖς ἴσας ποιῶσιν, ἐπʼ εὐθείας ἔσονται ἀλλήλαις αἱ εὐθεῖαι.

If, at any straight line and at a point on it, two straight lines not lying on the same side make the adjacent angles equal to two right angles, the straight lines will be in a straight line with one another.

πρὸς γάρ τινι εὐθείᾳ τῇ ΑΒ καὶ τῷ πρὸς αὐτῇ σημείῳ τῷ Β δύο εὐθεῖαι αἱ ΒΓ, ΒΔ μὴ ἐπὶ τὰ αὐτὰ μέρη κείμεναι τὰς ἐφεξῆς γωνίας τὰς ὑπὸ ΑΒΓ, ΑΒΔ δύο ὀρθαῖς ἴσας ποιείτωσαν· λέγω, ὅτι ἐπʼ εὐθείας ἐστὶ τῇ ΓΒ ἡ ΒΔ. εἰ γὰρ μή ἐστι τῇ ΒΓ ἐπʼ εὐθείας ἡ ΒΔ, ἔστω τῇ ΓΒ ἐπʼ εὐθείας ἡ ΒΕ. ἐπεὶ οὖν εὐθεῖα ἡ ΑΒ ἐπʼ εὐθεῖαν τὴν ΓΒΕ ἐφέστηκεν, αἱ ἄρα ὑπὸ ΑΒΓ, ΑΒΕ γωνίαι δύο ὀρθαῖς ἴσαι εἰσίν· εἰσὶ δὲ καὶ αἱ ὑπὸ ΑΒΓ, ΑΒΔ δύο ὀρθαῖς ἴσαι· αἱ ἄρα ὑπὸ ΓΒΑ, ΑΒΕ ταῖς ὑπὸ ΓΒΑ, ΑΒΔ ἴσαι εἰσίν.

For, at some straight line AB and at the point B on it, let two straight lines BC, BD not lying on the same side make the adjacent angles ABC, ABD equal to two right angles; I say that BD is in a straight line with CB. For if BD is not in a straight line with BC, let BE be in a straight line with CB. Since, therefore, the straight line AB stands on the straight line CBE, the angles ABC, ABE are therefore equal to two right angles; and the angles ABC, ABD are also equal to two right angles; therefore the angles CBA, ABE are equal to the angles CBA, ABD.

κοινὴ ἀφῃρήσθω ἡ ὑπὸ ΓΒΑ· λοιπὴ ἄρα ἡ ὑπὸ ΑΒΕ λοιπῇ τῇ ὑπὸ ΑΒΔ ἐστιν ἴση, ἡ ἐλάσσων τῇ μείζονι· ὅπερ ἐστὶν ἀδύνατον.

Let the common angle CBA be subtracted; therefore the remaining angle ABE is equal to the remaining angle ABD, the less to the greater; which is impossible.

οὐκ ἄρα ἐπʼ εὐθείας ἐστὶν ἡ ΒΕ τῇ ΓΒ. ὁμοίως δὴ δείξομεν, ὅτι οὐδὲ ἄλλη τις πλὴν τῆς ΒΔ· ἐπʼ εὐθείας ἄρα ἐστὶν ἡ ΓΒ τῇ ΒΔ. ἐὰν ἄρα πρός τινι εὐθείᾳ καὶ τῷ πρὸς αὐτῇ σημείῳ δύο εὐθεῖαι μὴ ἐπὶ τὰ αὐτὰ μέρη κείμεναι τὰς ἐφεξῆς γωνίας δυσὶν ὀρθαῖς ἴσας ποιῶσιν, ἐπʼ εὐθείας ἔσονται ἀλλήλαις αἱ εὐθεῖαι· ὅπερ ἔδει δεῖξαι.

Therefore, BE is not in a straight line with CB. Similarly, we can prove that neither is any other straight line except BD; therefore CB is in a straight line with BD. Therefore, if, at any straight line and at a point on it, two straight lines not lying on the same side make the adjacent angles equal to two right angles, the straight lines will be in a straight line with one another. - Being what it was required to prove.

§1.prop.15ἐὰν δύο εὐθεῖαι τέμνωσιν ἀλλήλας, τὰς κατὰ κορυφὴν γωνίας ἴσας ἀλλήλαις ποιοῦσιν.

If two straight lines cut one another, they make the vertical angles equal to one another.

δύο γὰρ εὐθεῖαι αἱ ΑΒ, ΓΔ τεμνέτωσαν ἀλλήλας κατὰ τὸ Ε σημεῖον· λέγω, ὅτι ἴση ἐστὶν ἡ μὲν ὑπὸ ΑΕΓ γωνία τῇ ὑπὸ ΔΕΒ, ἡ δὲ ὑπὸ ΓΕΒ τῇ ὑπὸ ΑΕΔ. ἐπεὶ γὰρ εὐθεῖα ἡ ΑΕ ἐπʼ εὐθεῖαν τὴν ΓΔ ἐφέστηκε γωνίας ποιοῦσα τὰς ὑπὸ ΓΕΑ, ΑΕΔ, αἱ ἄρα ὑπὸ ΓΕΑ, ΑΕΔ γωνίαι δυσὶν ὀρθαῖς ἴσαι εἰσίν.

For let two straight lines AB, CD cut one another at the point E; I say that the angle AEC is equal to the angle DEB, and the angle CEB to the angle AED. For since the straight line AE stands on the straight line CD, making the angles CEA, AED, the angles CEA, AED are therefore equal to two right angles.

πάλιν, ἐπεὶ εὐθεῖα ἡ ΔΕ ἐπʼ εὐθεῖαν τὴν ΑΒ ἐφέστηκε γωνίας ποιοῦσα τὰς ὑπὸ ΑΕΔ, ΔΕΒ, αἱ ἄρα ὑπὸ ΑΕΔ, ΔΕΒ γωνίαι δυσὶν ὀρθαῖς ἴσαι εἰσίν.

Again, since the straight line DE stands on the straight line AB, making the angles AED, DEB, the angles AED, DEB are therefore equal to two right angles.

ἐδείχθησαν δὲ καὶ αἱ ὑπὸ ΓΕΑ, ΑΕΔ δυσὶν ὀρθαῖς ἴσαι· αἱ ἄρα ὑπὸ ΓΕΑ, ΑΕΔ ταῖς ὑπὸ ΑΕΔ, ΔΕΒ ἴσαι εἰσίν.

And the angles CEA, AED were also proved equal to two right angles; therefore the angles CEA, AED are equal to the angles AED, DEB.

κοινὴ ἀφῃρήσθω ἡ ὑπὸ ΑΕΔ· λοιπὴ ἄρα ἡ ὑπὸ ΓΕΑ λοιπῇ τῇ ὑπὸ ΒΕΔ ἴση ἐστίν·

Let the common angle AED be subtracted; therefore the remaining angle CEA is equal to the remaining angle BED.

ὁμοίως δὴ δειχθήσεται, ὅτι καὶ αἱ ὑπὸ ΓΕΒ, ΔΕΑ ἴσαι εἰσίν.

Similarly, it can be proved that the angles CEB, DEA are also equal.

ἐὰν ἄρα δύο εὐθεῖαι τέμνωσιν ἀλλήλας, τὰς κατὰ κορυφὴν γωνίας ἴσας ἀλλήλαις ποιοῦσιν· ὅπερ ἔδει δεῖξαι.

Therefore, if two straight lines cut one another, they make the vertical angles equal to one another. - Being what it was required to prove.

Notes

1.prop.14ἐπʼ εὐθείας ἔσονται ἀλλήλαις — In the phrase ἐπʼ εὐθείας ἔσονται ἀλλήλαις αἱ εὐθεῖαι, the dative ἀλλήλαις ('to one another') depends on the idiom ἐπʼ εὐθείας εἶναί τινι ('to be in a straight line with something'), indicating that the two lines form a single continuous straight line with respect to each other.
1.prop.14ἡ ἐλάσσων τῇ μείζονι· ὅπερ ἐστὶν ἀδύνατον. — An elliptical expression where a predicate of equality like ἴση ἐστίν ('is equal') is omitted: '[the] less [being equal] to the greater, which is impossible.' This points to a contradiction based on Common Notion 5 ('the whole is greater than the part') within the reductio ad absurdum.
1.prop.15τὰς κατὰ κορυφὴν γωνίας — A technical geometrical term literally meaning 'the angles according to the vertex,' which refers to the 'vertical' or 'opposite' angles. The prepositional phrase κατὰ κορυφήν modifies the noun γωνίας attributively.

Cite this passage

Euclid, Elements §1.prop.14-1.prop.15. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:1.prop.14-1.prop.15

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