1. A proof that can be proved in Euclidean geometry, yet not in Non-Euclidean geometry.

2. The sum of the interior angles of a triangle makes two right angles

3. All of these triangles have interior angles summing to two right angles, or 180 degrees, or π radians.

4. For any triangle, the sum of the interior angles is always equal to two right angles. We will prove that angle ABC plus angle BCA plus angle CAB is equal to two right angles.

5. Draw CE, parallel to AB. Extend line BC to a point D.

6.  CAB =  ACE  If two parallel lines are cut by a transversal, the alternate interior angles are equal.  ABC =  ECD  If two parallel lines are cut by a transversal, the corresponding angles are equal.

7.  ABC +  CAB =  ACE +  ECD  Equals added to equals are equal.  ABC +  CAB =  ACD   ACE +  ECD=  ACD

8.  ABC +  CAB +  BCA =  ACD +  BCA  Add  BCA. Equals added to equals are equal.  ACD +  BCA = Two right angles  Two angles on a straight line are either two right angles, or equal to two right angles.

9.  ABC+  CAB +  BCA is equal to two right angles.  Things which equal the same thing are also equal to one another.

10. Why can’t this theorem be proved in Non-Euclidean geometry? The sum of the interior angles of a triangle equals two right triangles.

11. The key is in the justification of these two steps in our proof. We need to use the parallel postulate to prove each of these statements. If two parallel lines are cut by a transversal, the alternate interior angles are equal. If two parallel lines are cut by a transversal, the corresponding angles are equal.

12. Parallel Postulate: If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. If  AEF +  CFE is less than two right angles, lines AB and CD will surely meet.

13. By denying the parallel postulate, we can create many new (Non Euclidean) geometries.

14. This triangle, on the hyperbolic plane, has interior angles adding to one right angle.

15. This triangle, on the elliptic plane, has interior angles summing to 232 degrees.

16. In this diagram, the line AB has two parallel lines passing through the point C.

17. References T. L. Heath (1956). Euclid’s Elements. New York: Dover. Illustrations created with Cinderella.