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Through any two points there is exactly one line
postulate
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through any three noncollinear points there is exactly one plane containing them
postulate
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If two points lie in a plane, then the line containing those points lies in the plane
postulate
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if two lines intersect, then they intersect in exactly one point
postulate
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if two planes intersect, then they intersect in exactly one line
postulate
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the points on a line can be put into a one-to-one correspondence with the real numbers
ruler postulate
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If B is between A and C, then AB+BC=AC
Segment Addition Postulate
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given line AB and a point O on line AB, all rays that can be drawn from O can be put into a one-to-one correspondence with the real numbers from 0 to 180
protractor postulate help classify angles by their measure
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If S is in the interior of <PQR, then m<PQS + m<SQR= m<PQR
< Add. Post.
angle addition postulate
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if p --> q is a true statement and p is true, then q is true
law of detachment
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if p --> and q --> r are true statements, then p --> r is a true statement
law of syllogism
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addition property of equality
if a=b, then a+c=b+c
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subtraction property of equality
if a=b, then a-c =b-c
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multiplication property of equality
if a=b, then ac=bc
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division property of equality
if a=b, and c DOES NOT = 0, then a/c = b/c
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reflexive property of equality
a=a
duhhh
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symmetric property of equality
if a=b then b=a
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transitive property of equality
if a=b and b=c, then a=c
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substitution property of equality
if a=b, then b can be substituted for a in any expression
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reflexive property of congruence
figure a ~= figure a
line EF =~ line EF
Reflex. Prop. of =~
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Symmetric property of congruence
if figure a ~= figure B, then figure B ~= figure A
sym. prop. of ~=
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Transitive Property of Congruence
If figure A~= figure B and figure B ~= figure C, than figure A~= figure C
trans. prop. of ~=
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linear pair theorem
if two angles form a linear pair, then they are supplementary.
hypothesis: <A and <B form a linear pair
Conclusion: <a and <b are supplementary
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congruent supplements theorem
if two angles are supplementary to the same angle *or to two congruent sides), then the two angles are congruent
hypothesis: <1 and <2 are supplementary. <2 and <3 are supplementary
conclusion: <1 ~= <3
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right angle congruence theorem
all right angles are congruent
Hypothesis: <a and <b are right angles
conclusionL <a~=<b
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congruent complements theorem
if two angles are complementary to the same angle (or to 2 congruent <'s) then the 2 <'s are congruent
Hypothesis: <1 and <2 are complementary. <2 and <3 are complementary
conclusion: <1~=<3
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common segments theorem
given collinear points A, B, C, and D arranged, if line AB ~= line CD, then line AC~= line BD
Hypothesis: line AB ~= line CD
Conclusion: line AC ~= line BD
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Vertical Angles Theorem
Vertical angles are congruent
Hypothesis: <a and <b are vertical angles
conclusion: <a ~= <b
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if two congruent angles are supplementary, then each angle is a right angle
~= <'s suppl. --> right <'s
- hypothesis:
- <1~= <2 <1 and <2 are supplementary
conclusion: <1 and <2 are right angles
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corresponding angles postulate
if two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent
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alternate interior angles theorem
if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent
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alternate exterior angles theorem
if 2 || lines are cut by a transversal, then the pairs of alternate exterior angles are congruent
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same-side interior angles theorem
if 2 || lines are cut by a transversal, then the 2 pairs of same-side interior angles are supplementary
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converse of the corresponding angles postulate
if 2 coplanar lines are cut by a transversal so that a pair of corresponding angles are congruent, then the 2 lines = ||
Hypothesis: <1 ~= <2
conclusion: m || n
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parallel postulate
through a point P not on line l, there is exactly one line parallel to l
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converse of the alternate interior angles theorem
if 2 coplanar lines are cut by a transversal so that a pair of alternate interior angles are congruent, then the 2 lines are parallel
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converse of the alternate exterior angles theorem
if 2 coplanar lines are cut by a transversal so that a pair of alternate exterior angles are congruent, then the 2 lines are parallel
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converse of the same side interior angles theorem
if 2 coplanar lines are cut by a transversal so that a pair of same side interior angles are supplementary, the the 2 lines are parallel
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2 intersecting lines form a linear pair of congruent angles --> the lines are perpendicular
theorem
conclusion: l is perpendicular to m
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perpendicular transversal theorem
in a plane, if a transversal is perpendicular to one of 2 parallel lines, then it is perpendicular to the other line
hypothesis: transversal perpendicular to one parallel line
conclusion: transversal perpendicular to the other parallel line
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2 lines perpendicular to the same line --> the 2 lines are ||
theorem
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parallel lines theorem
in a coordinate plane, 2 nonvertical lines are parallel iff they have the same slope.
any 2 vertical lines are parallel
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perpendicular lines theorem
in a coordinate plane, 2 non vertical lines are perpendicular iff the product of their slopes is -1
vertical and horizontal lines are perpendicular
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triangle sum theorem
the sum of the angle measures of a triangle is 180 degrees
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the acute angles of a right triangle are complementary
corollary
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the measure of each angle of an equiangular triangle is 60 degrees
corollary
m<a = m<b = m<c = 60 degrees
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exterior angle theorem
the measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles
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third angles theorem
if 2 angles of one triangle are congruent to 2 angles of another triangle ---> third pair of angles are congruent
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side side side (SSS) congruence postulate
if 3 sides of 1 triangle are congruent to 3 sides of another triangle, then the triangles are congruent
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side angle side (SAS) congruence
if 2 sides and the included angle of 1 triangle are congruent to 2 sides and the included angle of another triangle --> the triangles are congruent
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angle side angle (ASA) congruence
2 angles and the included side of 1 triangle are congruent to 2 angles and the included side of another triangle --> congruent triangles
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angle angle side (AAS) congruence theorem
2 angles and non included side of 1 triangle are ~= to the corresponding angles and non included side of another triangle --> triangles are congruent
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Hypotenuse-leg (HL) congruence
the hypotenuse and a leg of a right triangle are ~= to the hypotenuse and a leg of another right triangle --> the triangles are congruent
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CPCTC
corresponding parts of congruent triangles are congruent
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isosceles triangle theorem
2 sides of a triangle are congruent --> angles opposite those angles are congruent
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equilateral triangle corollary
if a triangle is equilateral --> it's equiangular
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equiangular triangle corollary
equiangular triangle --> equilateral triangle
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perpendicular bisector theorem
point is on the perpendicular bisector of a segment --> it is equidistant from the endpoints of the segment
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converse of the perpendicular bisector theorem
a point is equidistant from the endpoints of a segment ---> perpendicular bisector of the segment
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angle bisector theorem
if a point is on the bisector of an angle --> equidistant from the sides of the angle
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converse of the angle bisector theorem
if a point in the interior of an angle is equidistant from the sides of the angle --> on the bisector of the angle
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circumcenter theorem
the circumcenter of a triangle is equidistant from the vertices of the triangle
can be inside the triangle, outside the triangle, or on the triangle
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incenter theorem
the incenter of a triangle is equidistant from the sides of the triangle
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centroid theorem
the centroid of a triangle is located 2/3 of the distance from each vertex to the midpoint of the opposite side
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triangle midsegment theorem
a midsegment of a triangle is || to a side of the triangle, and it's length is half the length of that side.
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if 2 sides of a triangle are not congruent --> larger angle is opposite the longer side
theorem
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in triangle, the longer side is oposite the larger angle
theorem
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triangle inequality theorem
the sum of any 2 side lengths of a triangle is greater than the 3rd side length
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