Question

8. True or false (in absolute geometry unless otherwise stated.) (a) If A and D are points on opposite sides of BC and LABC BCD, then AB II CD (b) If two lines are parallel, then they are equidistant from each other. (c) If oABCD is a quadrilateral with right angles at A, B, and C, then LD is also a right angle. (d) Euclid's Parallel Postulate is equivalent to the following statement: Every point in the interior of an angle lies on a line intersecting both sides of the angle not at the vertex (e) Euclid's Parallel Postulate is equivalent to the following statement: Given a point P that is not on a line I, there is a unique line through P perpendicular to (f) There exists a plane satisfying the axioms of absolute geometry as well as the Hypothesis of the Obtuse Angle. (g) There exists a plane satisfying the axioms of absolute geometry as well as the Hypothesis of the Acute Angle

Answer 1

(a) true

since the angles are alternate interior angles,

ABC and〈B C D

"if alternate interior angles are congruent then the lines are parallel"

so AB and CD are parallel

(b)true

since the converse to equidistance theorem states that "if two lines are parallel then each one is equidistant from the other"

(c)true

since this statement is equivalent to euclid's parallel postulate ,which hold in absolute geometry that "if three angles of a quadrilateral are right angles the so is fourth"

(d) true

since its an equivalent to parallel postulate with in the absolute geometry which says "through any point in the interior of an angle there exists a line which intersects both arms of the angle ;not at the vertex