Question

1.      This exercise asks you to negate various results that are equivalent to the Euclidean parallel postulate.

(a)   Alternate Postulate 5.1. Given a line and a point not on the line, exactly one line can be drawn through the given point and parallel to the given line.

(b)   Alternate Postulate 5.2. If two parallel lines are cut but a transversal, then the alternate interior angles are equal, each exterior angle is equal to the opposite interior angle, and sum of the interior angles on the same side of the transversal is 180⁰.

(c)   Alternate Postulate 5.4. If a straight line intersects one of two parallel lines, it will also intersect the other.

(d)   Alternate Postulate 5.9. [Saccheri 3 ( Euclid’s Proposition I.32 )] The angle sum of any triangle is 180⁰.

(e)   Alternate Postulate 5.11 [Pythagorean Theorem] In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.

(f)    Alternate Postulate 5.18. [Farkas Bolyai, 1800] Given three noncollinear points, there is a circle passing through all three.

Answer 1

1. There exist some line A and some point Q not on the line A such that there doesnot exist a line through Q and parallel to A.
2. Two lines are cut by a transversal and alternate interior angles are not equal;each exterior angle is not equal to the opposite interior angles and some of the interior angles on same side of the transversal is not 180 degree
3. A straight line intersects one of two parallel lines and doesn't intersects other
4. The angle sum of triangle is not 180 degree at all.