Theorem

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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

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abisector of the segment. If M is the midpoint of XY, then line k, plane Z, line

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MR and line MT all bisect XY .

Theorem 1.5Midpoint Theorem If M is the midpoint of a segment AB, then

2AM=AB 2MB=AB

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.

Theorem 1.8Angle Bisector Theorem If ray OX is a bisector of <AOB, then

2m<AOX=2m<AOB 2m<XOB=m<AOB

m<AOX=1/2m<AOB and m<XOB=1/2m<AOB

Theorem 1.9 If two angles are vertical, then they are congruent.

Since lines l and m intersect, vertical angles are formed. Angles 1 and 3 form one pair of vertical angles.

1 and 2 form a linear pair. 3 and 2 form a linear pair.

m 1 + m 2 = 180. m 3 + m 2 = 180

m 1 = 180 - m 2 m 3 = 180 - m 2

Therefore, 1 and 3 are equal in measure and congruent.

Given: A and C are vertical angles

Prove: A ≅ C

Proof:

Statements Reasons

1. A and C are vertical angles 1. Given

2. A and B form a linear pair 2. Definition of linear pair

3. C and B form a linear pair 3. Definition of linear pair

4. A and B are supplementary 4. Linear Pair postulate

5. C and B are supplementary 5. Linear pir postulate

6. A ≅ C6. Supplements of the same

angle are congruent

Theorem 1.10 If two lines are perpendicular, then the pairs of adjacent angles they form are congruent.

Theorem 1.11 If two lines intersect to form a pair of congruent adjacent angles, then the lines are perpendicular.

Theorem 1.12If there is given any point on a line in a plane, then there is exactly one line in that plane perpendicular to the given line at the given point.

Line AB lies in a plane R and contains P.

There exists line CD in R such that line CD contains P and line CD perpendicular to line AB. ( existence)

Only one line, line CD in R is perpendicular to line AB at P. ( uniqueness)

Theorem 1.13 If the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary.

Theorem 1.14 If there is a point not on a line, then there is exactly one line perpendicular to the given line through the given point.

Theorem 2.1 Congruence of segments is reflexive, symmetric and transitive.

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AB = CD and CD = EF, then AB = EF

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1. AB = CD and CD = EF 1. Given

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2. AB = CD and CD =

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