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54 Cards in this Set
- Front
- Back
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Reflexive property
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a = a
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Substitution property
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If a = b, then a can be substituted for b in any expression.
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Addition property of equality
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If a = b
then a + c = b + c |
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Subtraction property of equality.
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If a = b
then a - c = b - c |
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Multiplication property of equality.
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If a = b
then ac = bc |
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The division property of equality.
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If a = b
then a ÷ c = b ÷ c |
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Every point on a line corresponds to exactly one real number called its...
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coordinate.
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To every pair of points on a line there corresponds a real number called the...
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distance.
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The Ruler Postulate
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The points on a line can be numbered so that positive number differences measure distances.
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A-B-C
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states betweenness, and is read “B is between A and C”.
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Betweenness of points
A - B - C iff a < b < c or c < b < a |
A point is between two other points on the same line iff its coordinate is between their coordinates.
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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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A unit for measuring angles.
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Number of degrees in a circle.
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360
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A half-rotation of rays...
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is all of the rays that correspond to a common protractor.
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The rays in a half-rotation can be numbered...
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so that to every ray there corresponds exactly one real number called its coordinate
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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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An angle is acute...
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...iff it is less than 90°.
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An angle is right...
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...iff it is 90°.
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An angle is obtuse...
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...iff it is more than 90° but less than 180°.
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An angle is straight...
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...iff it is 180°.
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Betweenness of Rays
rayOA - rayOB - rayOC iff a < b < c or c < b < a |
A ray is between two others in the same half-rotation iff its coordinate is between their coordinates.
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Betweenness of Rays Theorem
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If rayOA - rayOB - rayOC,
then ∠AOB + ∠BOC = ∠AOC |
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Definition of midpoint of a line segment.
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A point is the midpoint of a line segment iff it divides the line segment into two equal segments.
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Definition of angle bisector.
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A line bisects an angle iff it divides the angle into two equal angles.
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congruent
(‘con-GREW-ent’) |
Informally, same size, same shape, and referencing polygons or more complex figures. Line segments or angles the same size are ‘equal’, but triangles may be congruent.
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corollary
(‘COR-oh-lair-ee’) |
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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Two angles are complementary...
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...iff their sum is 90°
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Two angles are supplementary...
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...iff their sum is 180°
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Complementary angle theorem.
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Complements of the same angle are equal.
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Supplementary angle theorem.
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Supplements of the same angle are equal.
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Two angles are a linear pair...
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...iff they have a common side and their other sides are opposite rays.
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Two angles are vertical angles...
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...iff the sides of one angle are opposite rays to the sides of the other.
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Theorem: The angles in a linear pair ...
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...are supplementary angles.
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Theorem: Vertical angles ...
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... are equal angles.
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Two lines are perpendicular...
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... iff they form a right angle.
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Theorem: Perpendicular lines form ...
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... 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: If the angles in a linear pair are equal …
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... then their sides are perpendicular.
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Two lines are parallel ...
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... iff they lie in the same plane and do not intersect.
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Complementary or supplementary angles do not need to...
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share a side or a vertex. They may be disconnected.
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If an angle is less than 90°...
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then it is acute.
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If an angle is 90° ...
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then it is right.
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If an angle is greater than 90° but less than 180°...
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then it is obtuse.
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If two lines lie in the same plane but do not intersect...
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then they are parallel.
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If two lines form a right angle...
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then they are perpendicular.
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If the sides of one angle are opposite rays to the sides of the other...
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then they are vertical angles.
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If two angles have a common side and their other sides are opposite rays...
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then they are a linear pair.
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If two angles add to 180°...
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then they are supplementary.
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If two angles add to 90°...
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then they are complementary.
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If a line bisects an angle into two equal angles...
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then it is an angle bisector.
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If a point divides a line segment into two equal segments...
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then it is the midpoint.
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