Proposition 27

Theorem

If a line (EF) intersecting two lines (AB, CD) makes the alternate angles equal to each other (∠AEF = ∠EFD), then the two lines are parallel (AB ‖ CD).

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Commentary

Let AB and CD be two given lines and let EF be a line intersecting these two lines. Let the two alternate angles, ∠AEF and ∠EFD made by the intersection be equal.

Then the two lines AB and CD are parallel.

Euclid's words "a line falls on two other lines" means that the first line is intersecting the other two.

This proposition is conceptually related to postulate 5 and is the converse of one of the conclusions of Book 1 Proposition 29.

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Computable version

In[1]:=

GeometricScene[{{A.,B.,C.,O.,D.,E.,F.,G.},{}},​​{GeometricStep[{CircleThrough[{A.,B.,C.},O.],Line[{A.,O.,C.}],Line[{D.,E.}],EuclideanDistance[D.,E.]≤EuclideanDistance[A.,C.]}],​​GeometricStep[{F.∈Line[{A.,C.}],EuclideanDistance[A.,F.]EuclideanDistance[D.,E.]}],​​GeometricStep[{GeometricAssertion[{CircleThrough[{F.},A.],CircleThrough[{A.,B.,C.},O.]},{"Concurrent",G.}]}],​​GeometricStep[{Line[{A.,G.}]}]},​​{EuclideanDistance[A.,G.]EuclideanDistance[D.,E.]}​​]//RandomInstance

Out[]=

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Additional instances

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