Ἐπεὶ γὰρ εὐθεῖα ἡ ΑΕ ἐπ᾽ εὐθεῖαν τὴν ΓΔ ἐφέστηκε γωνίας ποιοῦσα τὰς ὑπὸ ΓΕΑ, ΑΕΔ, αἱ ἄρα ὑπὸ ΓΕΑ, ΑΕΔ γωνίαι δυσὶν ὀρθαῖς ἴσαι εἰσίν.
πάλιν, ἐπεὶ εὐθεῖα ἡ ΔΕ ἐπ᾽ εὐθεῖαν τὴν ΑΒ ἐφέστηκε γωνίας ποιοῦσα τὰς ὑπὸ ΑΕΔ, ΔΕΒ,
αἱ ἄρα ὑπὸ ΑΕΔ, ΔΕΒ γωνίαι δυσὶν ὀρθαῖς ἴσαι εἰσίν.
ἐδείχθησαν δὲ καὶ αἱ ὑπὸ ΓΕΑ, ΑΕΔ δυσὶν ὀρθαῖς ἴσαι:
αἱ ἄρα ὑπὸ ΓΕΑ, ΑΕΔ ταῖς ὑπὸ ΑΕΔ, ΔΕΒ ἴσαι εἰσίν.
κοινὴ ἀφῃρήσθω ἡ ὑπὸ ΑΕΔ:
λοιπὴ ἄρα ἡ ὑπὸ ΓΕΑ λοιπῇ τῇ ὑπὸ ΒΕΔ ἴση ἐστίν:
ὁμοίως δὴ δειχθήσεται, ὅτι καὶ αἱ ὑπὸ ΓΕΒ, ΔΕΑ ἴσαι εἰσίν.
For, since the straight line AE stands on the straight line CD, making the angles CEA, AED, the angles CEA, AED are equal to two right angles [I. 13]
Again, since the straight line DE stands on the straight line AB, making the angles AED, DEB,
the angles AED, DEB are equal to two right angles. [I. 13]
But the angles CEA, AED were also proved equal to two right angles;
therefore the angles CEA, AED are equal to the angles AED, DEB. [Post. 4 and C. N. 1]
Let the angle AED be subtracted from each;
therefore the remaining angle CEA is equal to the remaining angle BED. [C. N. 3]
Similarly it can be proved that the angles CEB, DEA are also equal.
Quoniam enim recta AE stat super rectam GD angulos faciens GEA et AED, anguli ergo GEA et AED duobus rectis sunt equales.
Rursus quoniam recta DE stat super rectam AB angulos faciens AED et DEB,
anguli ergo AED et DEB duobus rectis sunt equales.
Demonstrati vero sunt et GEA et AED duobus rectis equales.
Anguli ergo GEA et AED angulis AED et DEB sunt equales.
Communis auferatur AED,
reliquus ergo GEA reliquo BED equalis est.
Similiter autem ostendetur quoniam anguli GEB et DEA sunt equales.
برهامه ان خط (ا ه) قائم على خط (ج د) فببرهان يج من (ا) تكون زاويتا (ا ه ج) (ا ه د) معادلتين لقائتين
وايضا خط (ج ه) قائم على خط (ا ب)
فزاويتا (ا ه ج) (ج ه ب) معادلتان لزاويتين قائمتين
فننقص زاوية (ا ه ج) المشتركة فتبقى زاوية (ا ه د) مساوية لزاوية (ج ه ب)
ايضا فان خط (ج ه) قائم على خط (ا ب) فزاويتا (ا ه ج) (ج ه ب) معادلتان لزاويتين قائمتين
فنسقط زاوية (ج ه ب) المشتركة
فتبقى زاوية (ا ه ج) مساوية لزاوية (ب ه د)
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