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Properties and charts

1.  Linear Pair Property
If two angles form a linear pair, then they are supplementary.

Algebraic Properties of Equality
Let a, b, and c be real numbers or expressions representing real numbers.

Addition Property
If a = b, then a + c = b + c
Subtraction Property
If a = b, then a - c = b - c
Multiplication Property
If a = b, then ac = bc
Division Property
If a = b, then a/c = b/c
Substitution Property
If a = b, you may relace a with b in any true equation containing a and the resulting equation will still be true.

If equals are added to equals, then the wholes are equal.

Proof:
x - 4 = 12                               given statement of equality.
x - 4 + 4 = 12 + 4                  Addition property of equalities  
                                               (4's , equal numbers, are added to equals, the sides of the equation.)
x  = 16                                   Simplify
                                               (x is a variable representing a whole number, 16 is a whole number)

Equivalence Properties of Equalities

Reflexive Property
For any real number a, a = a.
Symmetric Property
For all real numbers a and b, if a =b, then b = a.
Transitive Property
For all real numbers a, b, and c, if a = b and b = c, then a = c.

Equivalence Properties of Congruence

Reflexive Property
Figure A is congruent to figure A
Symmetric Property
If figure a is congruent to figure b, then figure b is congruent to figure a.
Transitive Property
If figure a is congruent to figure b, and figure b is congruent to figure c, then figure a is congruent to figure c.





 Three-dimensional chart for determining octant positions.
+
-
0
x
front
back
y-z plane
y
right
left
x-z plane
z
top
bottom
x-y plane


Consider the point ( 3, 1, 7).  All three are positive; therefore the point is located in the top, front, right octant or the "first octant."

Consider the point (4, 0, -2).  The y coordinate is 0; therefore, the point lies on the x-z plane.

Consider the point (-8, -1, -5) All three are negative; therefore, the point is located in the bottom, back, left octant.