Proposition 28a

Theorem

If a line (EF) intersecting two lines (AB, CD) makes the exterior angle (∠EGB) equal to its corresponding interior angle (∠GHD), 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 at points G and H, respectively.

Let the exterior angle ∠EGB and its corresponding interior angle ∠GHD made by the intersecting lines be equal.

Then the two lines AB and CD are parallel.

Euclid specified two cases of angles either being equal or adding up to two right angles. This proposition is one case and Book 1 Proposition 28b is the other.

This proposition 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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