Theorems- statements that must be proven true by citing undefined terms, definitions, postulates, and previously proven theorems.
Theorem 1.1 If two distinct lines intersect, then they intersect in exactly one point.
Lines l and m intersect at K. If l and m were to intersect at a second point, then both would contain the same two points. By Postulate 2, that is impossible. Therefore, K is the only point of intersection for lines l and m.
Theorem 1.2 If there is a line and a point not in the line, then there is exactly one plane that contains them.
Let r and D represent the line and point of this theorem. Postulate 1 says that r has at least two distinct points such as F and g. points D,F, and G are non-collinear, so by Postulate 3 there is exactly one plane that contains them. Postulate 4 says that all the other points in r must be in this plane as well. Hence, this is the one plane that contains r and D.
Theorem 1.3 If two distinct lines intersect, then they lie in exactly one plane.
Lines k and m intersect in point P. Consider another point Q on k. From Theorem 1.2, it is known that exactly one plane contains both m and Q. Postulate 4 says that since k contains P and Q, k lies in the same plane as P and Q and hence in the same plane as m.
" Exactly one" in Theorem 1.3 involves existence and uniqueness statements:
There exists at least one plane that contains the intersecting lines.
There is only one plane that contains the intersecting lines.
The first statement is for the existence of the plane, and the second is for the uniqueness of the plane. " Exactly one" implies existence and uniqueness.
Theorem 1.4 On a ray, there is exactly one point that is at a given distance from the endpoint of the ray.
Any line, segment, ray, or plane that intersects a segment at its midpoint is called
abisector of the segment. If M is the midpoint of XY, then line k, plane Z, line
MR and line MT all bisect XY .
Theorem 1.5Midpoint Theorem If M is the midpoint of a segment AB, then
AM=1/2 AB and MB=1/2 AB
Theorem 1.6 In a half plane, through the endpoint of a ray lying in the edge of the half plane, there is exactly one other ray such that the angle formed by the two rays has a given measure between 0 and 180.
Theorem 1.7 All right angles are congruent.
Given three coplanar rays OA , OT, and OB , OT is between OA and OB if and only if m AOT + m TOB = m AOB. A ray is a bisector of an angle if and only if it divides the angle into two congruent angles, thus angles of equal measure. If OX bisects AOB, then m AOX = m XOB.