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a rule that is accepted without proof
postulate/axiom
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the real number that corresponds to a point
coordinate
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the absolute value of the difference of the coordinates of A & B
distance
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two angles that share a common vertex & side but have no common interior points
adjacent angles
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two adjacent angles whose noncommon sides are opposite rays.
linear pair
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two angles whose sides form two pairs of opposite rays
vertical angles
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a closed plane figure
polygon
-
in a polygon if no line that contains a side of the polygon contains a point in the interior of the polygon
convex
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a polygon that is not conves
concave
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the distance around a fugure
perimeter
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the distance around a circle
circumference
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the amount of surface covered by a figure
area
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an unproven statement based on observations
conjecture
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find a pattern in specific cases & then write a conjecture for the general case
inductive reasoning
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a specific case for which the conjecture is false
counterexample
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a logical statement that has 2 parts a hypothesis & a conclusion
conditional statement
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a statement that is the opposite of the original statement
negation
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uses facts definitions accepted properties & laws of logic
deductive reasoning
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a = b, then a + c = b + c
addition prop
-
-
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a=b & c does not = 0 then a/c = b/c
division prop
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a = b ac=bc
multiplication prop
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a logical arguement that shows a statement is true.
proof
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has numbered statements & corresponding reasons
two-column proof
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a statement that can be proven
theorem
-
all right angles are congruent
right angles congruence theorem
-
if two angles are supplementary to the same angle (or to congruent angles) then they are congurent
- congruent supplements theorem
- same for complements
-
if two angles form a linear pair then they are supplementary
LINEAR PAIR POSTULATE
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vertical angles are congruent
vertical angles congruence theorem
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two lines that do not intersect & are coplanar
parllel lines
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two lines that do not intersect & are not coplanar
skew lines
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is there if a line and a point not on the line, then there if exactly one line through the point parallel to the given line
parallel postulate
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if there is a line & a point not on the line, then there is exactly one line through the point perpendicular to the given line
perpendicular postulate
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a line that intersects two or more coplanar lines & different points
transversal
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if two lines are parallel to the same line, then they are parallel to each other
transitive property of parallel lines
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in a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope
slopes of parallel lines
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two nonvertical lines are perpendicular if & only if the product of their slopes id -1
slopes of perpendicular lines
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if a transversal is perpendicular to 1 of 2 parallel lines, then it is perpendicular to the other
perpendicular transversal theorem
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in a plane if two lines are perpendicular to the same line, then they are parallel to each other
lines perpendicular to a transversal theorem
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the original angles
interior angles
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the angles that form linear pairs w/ the interior angles are the
exterior angles
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the sum of the measures of the interior angles of a triangle is 180
triangle sum theorem
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the measure of an exterior angle of a triangle is = tot he sum of the measures of the two nonadjacent interior angles
exterior angle theorem
-
a statement that can be proved easily using the theorem
corollary to a theorem
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the acute angles of a right angle are complementary
corollary to the triangle sum theorem
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if two angles of one triangle are congruent to two angles of another triangle, then the 3rd angles are also congruent
third angles theorem
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if two sides of a triangle are congruent then the angles opposite them are congruent
base angles theorem
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if two angles of a triangle are congruent then the angles opposite them are congruent
converse of base angles theorem
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if a triangle if equilateral then it is equiangular
corollary to the base angles theorem
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if a triangle is equiangular, then it is equilateral
corollary to the converse of base angles theorem
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an operation that moves or changes a geometric figure in some way to produce a new figure
transformation
-
-
moves every point of a figure the same direction & distance
translation
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uses a line of reflection to create a mirror image of the original figure
reflection
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turns a fig. around a fixed point
rotation
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a segment that connects the midpoints of two sides of the triangle
midpoint of the triangle
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the segment connecting the midpoints of two sides of a triangle is parallel to the 3rd side & is half as long as that side
midsegment theorem
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a seg., ray, line, or plane that is perpendicular to a segment @ its midpoint
perpendicular bisector
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same distance form each point
equidistant
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in a plane if a point is on the perpendicular bisector of a seg. then it is equidistant from the endpoints of the seg.
perpendicular bisector theorm
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in a plane if a point is equidistant from the endpoints of a seg. then it is on the perpendicular bisector of the segment
converse of the perpendicular bisector theorem
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when three or more lines,rays, or segs. intersect in the same point
- concurrent
- the point is the point of concurrency
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the perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle
concurrency of perpendicular bisectors of a triangle
-
the point of concurrency of the 3 perpendicular bisectors
circumcenter
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if a point is on the bisector of an angle, then it is equidistant form the two sides of the angle
angle bisector theorem
-
if a point is in the interior of an angle & is equidistant from the sides of the angle then it lies on the bisector of the angle
converse of the angle bisector theorem
-
the angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle
concurrency of angle bisectors of a triangle
-
the point of concurrency of the 3 angle bisectors of a triangle
incenter
-
a segment from a vertex to the midpoint of the opposite side.
median of triangle
-
point of concurrency of the medians of a triangle
centroid
-
the medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side.
concurrency of medians of a triangle
-
the perpendicular segment from vertex to the opposite side or line that contains the opposite side
altitude of a triangle
-
the lines containing the altitudes of a triangle are concurrent
concurrency of altitudes of a triangle
-
the point of concurrency for ltitudes
orthocenter
-
the sum of the lengths of any 2 sides of a triangle is greater than the length of the 3rd side
triangle inequality theorem
-
if two sides of one triangle are congruent to two sides of another triangle & the includes angle of the 1st is larger than the included angle of the 2nd then the third side of the 1st is longer then the 3rd of the second
hinge theorem
-
if two sides of one triangle are congruent to two sides of another triangle & the third side of the first is longer than the third side of the second then the included angle of the first is larger than the included angle of the second
converse of the hinge theorem
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