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Postulates, Theorems, and Constructions

Chapter 1 Tools of Geometry

Postulate 1-1

Through any two points there is exactly one line. (p. 13)

Postulate 1-2

If two distinct lines intersect, then they intersect in exactly one point. (p. 13)

Postulate 1-3

If two distinct planes intersect, then they intersect in exactly one line. (p. 14)

Postulate 1-4

Through any three noncollinear points there is exactly one plane. (p. 15)

Postulate 1-5

Ruler Postulate

Every point on a line can be paired with a real number. This makes a one-to-one correspondence between the points on the line and the real numbers. (p. 20)

Postulate 1-6

Segment Addition Postulate

If three points A, B, and C are collinear and B is between A and C, then A B + B C = A C . eh b plus b c . equals eh c , . (p. 21)

Postulate 1-7

Protractor Postulate

Consider O B → o b vector and a point A on one side of O B → . o b vector , . Every ray of the form O A → o eh vector can be paired one to one with a real number from 0 to 180. (p. 28)

Postulate 1-8

Angle Addition Postulate

If point B is in the interior of ∠ A O C , angle eh o c comma then m ∠ A O B + m ∠ B O C = m ∠ A O C . m angle eh o b plus m angle b o c equals m angle eh o c . (p. 30)

Postulate 1-9

Linear Pair Postulate

If two angles form a linear pair, then they are supplementary. (p. 36)

The Midpoint Formulas

On a Number Line

The coordinate of the midpoint M of A B ¯ eh b bar is a + b 2 . fraction eh plus b , over 2 end fraction . .

In the Coordinate Plane

Given A B ¯ eh b bar where a ( x 1 , y 1 ) eh open , x sub 1 , comma . y sub 1 , close and B ( x 2 , y 2 ) , b open , x sub 2 , comma . y sub 2 , close comma the coordinates of the midpoint of A B ¯ eh b bar are M ( x 1 + x 2 2 , y 1 + y 2 2 ) . m open . fraction x sub 1 , plus , x sub 2 , over 2 end fraction . comma . fraction y sub 1 , plus , y sub 2 , over 2 end fraction . close . . (p. 50)

The Distance Formula

The distance between two points a ( x 1 , y 1 ) eh open , x sub 1 , comma . y sub 1 , close and B ( x 2 , y 2 ) b open , x sub 2 , comma . y sub 2 , close is d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 . d equals . square root of open , x sub 2 , minus , x sub 1 , close squared . plus . open , y sub 2 , minus , y sub 1 , close squared end root . . (p. 52)

• Proof on p. 497, Exercise 35

The Distance Formula (Three Dimensions)

In a three-dimensional coordinate system, the distance between two points ( x 1 , y 1 , z 1 ) open , x sub 1 , comma . y sub 1 , comma . z sub 1 , close and ( x 2 , y 2 , z 2 ) open , x sub 2 , comma . y sub 2 , comma . z sub 2 , close can be found with this extension of the Distance Formula.

d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 + ( z 2 − z 1 ) 2 d equals . square root of open , x sub 2 , minus , x sub 1 , close squared . plus . open , y sub 2 , minus , y sub 1 , close squared . plus . open , z sub 2 , minus , z sub 1 , close squared end root (p. 56)

Postulate 1-10

Area Addition Postulate

The area of a region is the sum of the areas of its nonoverlapping parts. (p. 63)

Chapter 2 Reasoning and Proof

Law of Detachment

If the hypothesis of a true conditional is true, then the conclusion is true. In symbolic form:

If p → q p rightwards arrow q is true and p is true, then q is true. (p. 106)

Law of Syllogism

If p → q p rightwards arrow q is true and q → r q rightwards arrow r is true, then p → r p rightwards arrow r is true. (p. 108)

Properties of Congruence

Reflexive Property

A B ¯ ≅ A B ¯ eh b bar , approximately equal to , eh b bar and ∠ A ≅ ∠ A angle eh approximately equal to angle eh

Symmetric Property

If A B ¯ ≅ C D ¯ , eh b bar , approximately equal to . c d bar , comma then C D ¯ ≅ A B ¯ . c d bar , approximately equal to . eh b bar , .

If ∠ A ≅ ∠ B , angle , eh approximately equal to , angle b comma then ∠ B ≅ ∠ A . angle , b approximately equal to , angle eh .

Transitive Property

If A B ¯ ≅ C D ¯ , eh b bar , approximately equal to . c d bar , comma and C D ¯ ≅ E F ¯ , c d bar , approximately equal to . e f bar , comma then A B ¯ ≅ E F ¯ . eh b bar , approximately equal to . e f bar , .

If ∠ A ≅ ∠ B , angle , eh approximately equal to , angle b comma and ∠ B ≅ ∠ C , angle , b approximately equal to , angle c comma then ∠ A ≅ ∠ C . angle , eh approximately equal to , angle c .

If ∠ B ≅ ∠ A , angle , b approximately equal to , angle eh comma and ∠ B ≅ ∠ C , angle , b approximately equal to , angle c comma then ∠ A ≅ ∠ C . angle , eh approximately equal to , angle c . (p. 114)

Theorem 2-1

Vertical Angles Theorem

Vertical angles are congruent. (p. 120)

• Proof on p. 121

Theorem 2-2

Congruent Supplements Theorem

If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent. (p. 122)

• Proof on p. 123, Problem 3

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