For exercises 1-4, state whether a set of axioms could be:

1. Complete but not independent.

Ans: Yes. Solution: Just add theorems of the system as new axioms.

2. Independent but not complete.

Ans: Yes. Solution: Remove some axioms from a complete and independent set of axioms.

3. Independent but not consistent.

Ans: Yes. Solution: Add the negation of a theorem or axiom to a consistent and independent set of axioms.

4. Consistent but not independent.

Ans: Yes. Solution: Add theorems of a consistent system as new axioms.

20. Prove Playfair's axiom assuming Euclid's fifth postulate in the original form.

Solution: Let P be a point and l a line not through P. Let m be a line through P which meets l at a point S (see diagram below). Label two points on l, one on either side of S, with T and T'. By construction, find a point U so that angle TSP is congruent to angle SPU. Consider the line UP. If line UP meets line l on the T side of m, say at point Q, then SPQ is a triangle and angle SPU is an exterior angle of this triangle. This is a contradiction, since an exterior angle must always be greater in measure than either of the opposite interior angles (and TSP would be one of these angles). Thus PU does not meet l on the T side of m. Now, working with the congruent supplementary angles, T'SP and SPU', we can reach the same contradiction if PU met l on the T' side of m. Thus, PU can not meet l and so is parallel to l and passes through P. (Notice that this part did not use Euclid's fifth postulate).

Now assume that there is a second line through P which is parallel to l (not equal to PU), say k. There are two cases to consider, either a portion of k lies in the angle SPU or a portion lies in SPU'. In the first case, since the sum of the angles T'SP and SPU is 180 degrees, the sum of the angles WPS and T'SP is less than 180 degrees. By Euclid's 5th Postulate, k must meet l on the T' side of m, a contradiction since k was assumed to be parallel to l. In the second case, since the sum of the angles TSP and SPU' is 180 degrees, the sum of the angles WPS and TSP is less than 180 degrees. By Euclid's 5th Postulate, k must meet l on the T side of m, a contradiction since k was assumed to be parallel to l. So, in either case we get a contradiction to the existence of a second parallel to l through P, proving Playfair's axiom.

21. Prove the original statement of Euclid's fifth postulate, assuming Playfair's axiom.

Solution: Let lines l and k be crossed by the transversal line m. Suppose that m meets k at the point P, which is not on l. Furthermore, we may assume that the sum of the angles SPW and TSP is less than 180 degrees (see diagram below). This implies that the angle W'PS is greater than angle TSP. As in the first part of exercise 1, we can construct a parallel PU to l through P. As the angle SPU is congruent to angle TSP, PU can not be the line k. By Playfair's axiom, k can not be parallel to l ( since PU is the unique parallel to l through P). If k met l on the side of m which does not contain T, then angle SPW would be an exterior angle of a triangle which contained an angle which is the supplement of angle TSP. This is a contradiction since SPW is smaller than the supplement of TSP. Thus, k and l must meet on the side of m which contains T.

In Exercises 22-27 reword each sentance, using the negation of the conclusion, so that it becomes a statement assumed to be true in non-Euclidean geometry.

22. If a straight line intersects one of two parallel lines, it will always intersect the other.

Solution: If a straight line intersects one of two parallel lines, it will not always intersect the other.

23. Straight lines parallel to the same straight line are always parallel to one another.

Solution:Straight lines parallel to the same straight line are sometimes parallel to one another .

24.There exists one triangle for which the sum of the measures of the angles is \pi radians.

Solution:There does not exist a triangle for which the sum of the measures of the angles is \pi radians.

25. There exists a pair of similar but noncongruent triangles.

Solution:There does not exist a pair of similar but noncongruent triangles.

26. There exists a pair of straight lines the same distance apart at every point.

Solution:There does not exist a pair of straight lines the same distance apart at every point.

27. It is always possible to pass a circle through three noncollinear points.

Solution:It is not always possible to pass a circle through three noncollinear points.

28. If a straight line intersects one of two parallel lines, it will always intersect the other. Prove that this is equivalent to Euclid's fifth postulate.

Solution:We will use the Playfair form of the Euclidean fifth postulate. Suppose that lines l and k are parallel and line m intersects k at a point P. If m does not intersect l, then m is parallel to l. This would mean that there are two parallels to l through P (m and k), contradicting Playfair's axiom. Thus, m must also intersect l.

Now, let P be a point and l a line not through P. As in the first part of exercise 20, we can construct a line parallel to l through P, call it k. Let m be any other line through P. Since it intersects one of two parallel lines (k), it must intersect the other (l) by assumption. Thus, there is only one line through P which is parallel to l.