Proposition 15

Theorem

If two lines (AB , CD ) intersect one another, the opposite angles are equal (∠CEA = ∠BED, ∠CEB = ∠AED).

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Commentary

Let AB  and CD  be two given line segments, and let E be the point where they intersect.

Then the two pairs of opposite angles, ∠CEA and ∠BED, ∠CEB and ∠AED, are equal.

To avoid confusion, we used the term "opposite angles" in the statement, while Euclid used the term "vertical angles" to describe the pairs of opposite angles formed by intersecting lines. "Vertical" here means that the angles lie opposite one another with respect to a shared vertex, not the usual sense of "up and down."

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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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Dependency graphs