Proposition 29
Theorem
If a line (EF ) intersects two parallel lines (AB ‖ CD ), then alternate angles are equal to one another (e.g., ∠AGH = ∠GHD ), exterior angles are equal to their corresponding interior angles (e.g., ∠EGB = ∠GHD ) and two interior angles on the same side sum to two right angles (e.g., ∠BGH + ∠GHD = 180° ).
Commentary
1. Let AB and CD be two parallel lines and let EF be a line intersecting them at points G and H, respectively. Three types of angle relationships follow.
2. First, alternate angles (such as∠AGH and ∠GHD ) are equal. (This is the converse of Book 1 Proposition 27.)
3. Second, any exterior angle and its corresponding interior angle (such as∠EGB and ∠GHD ) are equal. (This is the converse of Book 1 Proposition 28a.)
4. Third, any two interior angles on the same side ofEF (such as ∠BGH and ∠GHD ) add up to two right angles. (This is the converse of Book 1 Proposition 28b.)
5. This is the first proposition in Euclid's Elements that depends on Postulate 5 for its proof.
2. First, alternate angles (such as
3. Second, any exterior angle and its corresponding interior angle (such as
4. Third, any two interior angles on the same side of
5. This is the first proposition in Euclid's Elements that depends on Postulate 5 for its proof.
Original statement
ἡ ϵἰς τὰς παραλλήλους ϵὐθϵίας ϵὐθϵῖα ἐμπίπτουσα τάς τϵ ἐναλλὰξ γωνίας ἴσας ἀλλήλαις ποιϵῖ καὶ τὴν ἐκτὸς τῇ ἐντὸς καὶ ἀπϵναντίον ἴσην καὶ τὰς ἐντὸς καὶ ἐπὶ τὰ αὐτὰ μέρη δυσὶν ὀρθαῖς ἴσας.
English translation
A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior angles on the same side equal to two right angles.
