In any triangle, if one of the sides is produced, then the exterior angle is greater than either of the interior and opposite angles.

Let *ABC* be a triangle, and let one side of it *BC* be produced to *D.*

I say that the exterior angle *ACD* is greater than either of the interior and opposite angles *CBA* and *BAC.*

Bisect *AC* at *E.* Join *BE,* and

produce it in a straight line to *F.*

Make *EF* equal to *BE,* join *FC,*

and draw *AC* through to *G.*

Since *AE* equals *EC,* and *BE* equals *EF,* therefore the two sides *AE* and *EB* equal the two sides *CE* and *EF* respectively, and the angle *AEB* equals the angle *FEC,* for they are vertical angles. Therefore the base *AB* equals the base *FC,* the triangle *ABE* equals the triangle *CFE,* and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides. Therefore the angle *BAE* equals the angle *ECF.*

But the angle *ECD* is greater than the angle *ECF,* therefore the angle *ACD* is greater than the angle *BAE.*

Similarly, if *BC* is bisected, then the angle *BCG,* that is, the angle *ACD,* can also be proved to be greater than the angle *ABC.*

Therefore *in any triangle, if one of the sides is produced, then the exterior angle is greater than either of the interior and opposite angles.*

Q.E.D.

Plane elliptic geometry is closely related to spherical geometry, but it differs in that antipodal points on the sphere are identified. Thus, a “point” in an elliptic plane is a pair of antipodal points on the sphere. A “straight line” in an elliptic plane is an arc of great circle on the sphere. When a “straight line” is extended, its ends eventually meet so that, topologically, it becomes a circle. This is very different from Euclidean geometry since here the ends of a line never meet when extended.

The illustration on the right shows the stereographic projection of one hemisphere. Since only one hemisphere is displayed, each “point” is represented by one point except those “points” such as A “triangle” in elliptic geometry, such as Elliptic geometry satisfies some of the postulates of Euclidean geometry, but not all of them under all interpretations. Usually, I.Post.1, to draw a straight line from any point to any point, is interpreted to include the uniqueness of that line. But in elliptic geometry a completed “straight line” is topologically a circle so that any pair of points on it divide it into two arcs. Therefore, in elliptic geometry exactly two “straight lines” join any two given “points.” |

Also, I.Post.2, to produce a finite straight line continuously in a straight line, is sometimes interpreted to include the condition that its ends don’t meet when extended. Under that interpretation, elliptic geometry fails Postulate 2.

Elliptic geometry fails I.Post.5, the parallel postulate, as well, since any two “straight lines” in an elliptic plane meet. That is, any two great circles on the sphere meet at a pair of antipodal points.

Finally, a completed “straight line” in the elliptic plane does not divide the plane into two parts as infinite straight lines do in the Euclidean plane. A completed “straight line” in the elliptic plane is a great circle on the sphere. Any two “points” not on that “straight line” include two points in the same hemisphere, and they can be joined by an arc that doesn’t meet the great circle. Therefore two “points” lie on the same side of the completed “straight line.”

The proof of this particular proposition fails for elliptic geometry, and the statement of the proposition is false for elliptic geometry. In particular, the statement “the angle *ECD* is greater than the angle *ECF*” is not true of all triangles in elliptic geometry. The line *CF* need not be contained in the angle *ACD.* All the previous propositions do hold in elliptic geometry and some of the later propositions, too, but some need different proofs.