from a chapter on Neutral Geometry strengthening foundations

*from a chapter on Neutral Geometry, strengthening foundations of Euclid\'s geometry by presenting 13 axioms plus continuity principles to replace his first 4 postulates. The 13 axioms are essentially those of David Hilbert known as the Hilbert Plane.

True or False

1) If two triangles have the same angle they are congruent

2) Euclid\'s fourth postualte is a theorem in neutral geometry

3) Theorem 4.4 shows that Euclid\'s fifth postualte is a theorem in neutral geometry

4) The Saccheri-Legrende theorem tells us that some triangles exist that have angle sums less than 180 degrees and some triangles exist that have angle sums equal to 180 degrees

5) The alternate interior angle theorem states that if parallel lines are cut by a transversal, then alternates interior angles are congruent to each other

*Theorem 4.4: Euclid\'s fifth parallel postulate <=> Hilbert\'s Euclidean parallel postulate

Solution

1. If two triangles have the same angle They are congruent

2. False

3. True

4. Yes, There may be either cases i.e., There will be triangles whose angles sums are less than 180 Or equal to 180 degrees

5. True, If parellel lines are cut by a transversal, then alternates interior angles are congruent to each other.