Question

2) (i) State the converse of the Alternate Interior Angle Theorem in Neutral Geometry. (ii) Prove...

2) (i) State the converse of the Alternate Interior Angle Theorem in Neutral Geometry.

(ii) Prove that if the converse of the Alternate Interior Angle Theorem is true, then all

triangles have zero defect.

[Hint: For an arbitrary triangle, ABC, draw a line through C parallel to side AB. Justify

why you can do this.]

5) Consider the following statements:

I: If two triangles are congruent, then they have equal defect.

II: If two triangles are similar, then they have equal defect.

III: If two triangles have equal defect, then they are similar.

IV: If two triangles have equal defect, then they are congruent.

Identify which statements can be proved in Neutral Geometry and which cannot be

proved in Neutral Geometry. If a statement can be proved, provide a proof. If it cannot

be proved, state why not.

3) Let us build a geometry, S, using the three axioms of incidence geometry with one

additional axiom added:

Incidence Axiom 1:

For every point P and every point Q (P and Q not equal), there

exists a unique line,l

, incident with P and Q.

Incidence Axiom 2:

For every line l

there exist at least two distinct points incident with l.

Incidence Axiom 3:

There exist (at least) three distinct points with the property that no

line is incident with all three of them.

Axiom 4

: There exist at most four points.

(i) The elliptic parallel property is true, false or independent of the axioms of S.

Which one is it? Prove your answer. . [Recall that the elliptic parallel

property says: Given a line l and a point P not on l, there exists no line through P parallel to l.]

(ii) The Euclidean parallel property is true, false or independent of the axioms of S.

Which one is it? Prove your answer. [Recall that the Euclidean parallel

property says: Given a line l

and a point P not on

l, there exists

exactly one line through P parallel to l.]

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