Postulates And Theorems

Question 1

Postulate 1 (The Distance Postulate)

Answer 1

To every pair of different points, there corresponds a unique positive number.

Question 2

Postulate 2 (The Ruler Postulate)

Answer 2

The points of a line can be placed in correspondence with the real numbers in such a way that (1) to every point of the line, there corresponds exactly one real number; (2) to every real number, there corresponds exactly one point of the line; (3) the distance between any two points is the absolute value of the difference of the corresponding numbers.

Question 3

Postulate 3 (The Ruler Placement Postulate)

Answer 3

Given two points, P and Q of a line, the coordinate system can be chosen in such a way that the coordinate of P is zero, and the coordinate of Q is positive.

Question 4

Postulate 4 (The Line Postulate)

Answer 4

For every two different points, there is exactly one line that contains both points.

Question 5

Postulate 5 (The Plane-Space Postulate)

Answer 5

(a) Every plane contains at least three different non-collinear points. (b) Space contains at least four different non-coplanar points.

Question 6

Postulate 6 (The Flat Plane Postulate)

Answer 6

If two points of a line lie in a plane, then the line lies in the same plane.

Question 7

Postulate 7 (The Plane Postulate)

Answer 7

Any three points lie in at least one plane, and any three non-collinear points lie in exactly one plane.

Question 8

Postulate 8 (Intersection of Planes Postulate)

Answer 8

If two different planes intersect, then their intersection is a line.

Question 9

Postulate 9 (The Plane Separation Postulate)

Answer 9

Given a line and a plane not containing it, the points of the plane that do not lie on the line form two sets such that (1) each of the sets is convex (2) If P is in one of the sets, and Q is in the other, then the segment PQ intersects the line.

Question 10

Postulate 10 (The Space Separation Postulate)

Answer 10

The points of space that do not lie in a given plane form two sets, such that (1) each of the sets is convex (2) If P is in one of the sets, and Q is in the other, then the segment PQ intersects the plane.

Question 11

Theorem 3-1

Answer 11

If two different lines intersect, their intersection contains only one point.

Question 12

Theorem 3-2

Answer 12

If a line intersects a plane not containing it, then the intersection contains only one point.

Question 13

Theorem 3-3

Answer 13

Given a line and a point not on the line, there is exactly one plane containing both.

Question 14

Theorem 3-4

Answer 14

Given two intersecting lines, there is exactly one plane containing both.

Question 15

Convex

Answer 15

A set M is called convex if for every two points P and Q of the set, the entire segment PQ lies in M.