Proposition 28b

Theorem

If a line (EF) intersecting two lines (AB, CD) makes the sum of two interior angles (∠BGH, ∠GHD) on the same side equal to the sum of two right angles, 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 two interior angles ∠BGH and ∠GHD on the same side of EF add up to two right angles.

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 28a is the other. This proposition is a partial converse of Postulate 5.

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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