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47 Cards in this Set
- Front
- Back
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(algebraic) reflexive property of equality
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a=a. Any number is equal to itself.
algebraic axiom or postulate |
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(algebraic) addition is commutative
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a + b = b + a. The order in which two numbers are added doesn't matter.
algebraic axiom or postulate |
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(algebraic) distributive property
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a(b + c) = ab + ac
algebraic axiom or postulate |
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(algebraic) squaring a binomial
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(a + b)² = a² + 2ab + b²
algebraic theorem |
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(algebraic) factoring the difference of two squares
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a² - b² = (a + b)(a - b)
algebraic theorem |
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The Reflexive Property
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a = a
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The Substitution Property
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If a = b, then a can be substituted for b in any expression.
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The Addition Property
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If a = b, then a + c = b + c.
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The Subtraction Property
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If a = b, then a - c = b - c.
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The Multiplication Property
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If a = b, then ac = bc.
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The Division Property
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If a = b and c ≠ 0, then a/c = b/c.
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Two points determine a ....
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a line is determined by these
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Coordinate
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exactly one real number that corresponds to a point on the number line
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Distance
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intuitive idea of what we measure by using a ruler. See Ruler Postulate...
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Postulate 3. The Ruler Postulate
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The points on a line can be numbered so that positive number distances measure distances.
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Definition, Betweenness of Points
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A point is between two other points on the same line iff its coordinate is between their coordinates. (More briefly, A-B-C iff a<b<c or a>b>c.
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Theorem 1. The Betweenness of Points Theorem
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If A-B-C, then AB + BC = AC
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degree
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the scale on a circle protractor is divided into 360 equal parts, each part measuring 1 degree.
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rotation of rays
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all the positions through which a ray with its endpoint at the center of a protractor moves as it turns exactly one around the scale
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Postulate 4. The Protractor Postulate
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The rays in a half-rotation can be numbered from 0 to 180 so that positive number differences measure angles.
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Definition acute angle
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iff it is less than 90 degrees
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Definition right angle
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an angle is right iff it is 90 degrees
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Definition obtuse angle
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an angle is obtuse iff it is more than 90 degrees but less than 180 degrees
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Definition straight angle
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an angle is straight iff it is 180 degrees
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definition Betweenness of Rays
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A ray is between two others in the same half-rotation iff its coordinate is between their coordinates. More briefly, OA-OB-OC iff a<b<c or a>b>c
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Theorem 2. The Betweenness of Rays Theorem
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If OA-OB-OC, then ∠AOB + ∠BOC = ∠AOC
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define midpoint of a line segment
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a point is a midpoint of a line segment iff it divides the line segment into two equal segments
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define bisect an angle
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a line bisects an angle iff it divides the angle into two equal angles
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congruent lines
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have equal lengths
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congruent angles
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have equal measures
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corollary
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a corollary is a theorem that can be easily proved as a consequence of a postulate or another theorem
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Corollary to the Ruler Postulate
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A line segment has exactly one midpoint.
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Corollary to the Protractor Postulate
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An angle has exactly one ray that bisects it
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definition complementary
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two angles are complementary iff their sum is 90 degrees
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definition supplementary
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two angles are supplementary iff their sum is 180 degrees
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Theorem 3. Complements of the same angle are equal.
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If ∠1 and ∠2 are complements of ∠3, then ∠1 = ∠2.
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Theorem 4. Supplements of the same angle are equal.
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If ∠1 and ∠2 are supplements of ∠3, then ∠1 = ∠2.
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definition opposite rays
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Rays AB and AC are opposite rays iff B-A-C.
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definition linear pair
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Two angles are a linear pair iff they have a common side and their other sides are opposite rays
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define vertical angles
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Two angles are vertical angles iff the sides of one angle are opposite rays to the sides of the other
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Theorem 5. The angles in a linear pair are supplementary
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If ∠1 and ∠2 are a linear pair, then ∠1 and ∠2 are supplementary.
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Theorem 6. Vertical angles are equal
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If ∠1 and ∠3 are vertical angles, then ∠1 = ∠3.
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define perpendicular
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Two lines are perpendicular iff they form a right angle
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Theorem 7. Perpendicular lines form four right angles
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If m ⊥ n, then m and n form four right angles
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Corollary to the definition of a right angle
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all right angles are equal
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Theorem 8. If the angles in a linear pair are equal, then their sides are perpendicular
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If ∠1 and ∠2 are a linear pair and ∠1 = ∠2, then OB ⊥ AC.
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define parallel
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two lines are parallel iff they lie in the same plane and do not intersect
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