Definitions, Postulates, Theorems for chapters 1-7 for Geometry – Flashcards
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the points on a line can be paired with the real numbers in such a way that any two points can have coordinates 0 and 1
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Ruler Postulate
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if B is between A and C, then: AB+BC=AC
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Segment Addition Postulate
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on line AB in a given plane, choose any point O between A and B. Consider ray OA and ray OB and all the rays that can be drawn from O on one side of line AB. These rays can be paired with the real numbers from 0 to 180 in such a way that: ray OA is paired with 0, and ray OB with 180, or, if ray OP is paired with X, and ray OQ with y, then angle POQ = l x-y l.
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Protractor Postulate
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If point B lies in the Interior of angle AOC, then angle AOB + angle BOC = angle AOC
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Angle Addition Postulate
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if angle AOC is a straight and and B is any point not on line AC, then angle AOB + angle BOC = 180
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Angle Addition Postulate
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a line contains at least two points; a plane contains at least three points not all in one line; space contains at least four points not all in one plane
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Postulate 5
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through any two points there is exactly one line
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Postulate 6
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through any three points there is at least one plane, and through any three non collinear points there is exactly one plane.
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postulate 7
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if two points are in a plane, then the line that contains the points is in that plane
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Postulate 8
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if two planes intersect, then their intersection is a line.
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Postulate 9
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If two lines intersect, then exactly one plane contains the lines
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Theorem 1-1
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through a line and a point not in the line there is exactly one plane
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Theorem 1-2
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If two lines intersect, then exactly one plane contains the lines
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Theorem 1-3
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if M is the midpoint of segment AB, then AM=1/2AB and MB=1/2AB
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Midpoint Theorem
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If ray BX is the bisector of angle ABC, then angle ABX = 1/2angleABC and angle XBC = 1/2 angle ABC
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Angle Bisector Theorem
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Reasons used in Proofs
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Given info, Definitions, Postulates (and properties from algebra), Proven Theorems
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Vertical angles are congruent
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Theorem 2-3
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Complementary angles
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two angles whose measures have the sum 90
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Supplementary angles
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two angles whose measures have the sum 180
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if two lines are perpendicular, then they from congruent adjacent angles
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Theorem 2-4
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if two lines form congruent adjacent angles, then the lines are perpindicular
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Theorem 2-5
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if the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary
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Theorem 2-6
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if two angles are supplements of congruent angles (or of the same angle) then the two angles are congruent
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Theorem 2-7
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if two angles are complements of congruent angles (or of the same angle), then the two angles are congruent
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Theorem 2-8
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Parallel lines
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coplanar lines that do not intersect
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Skew lines
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noncoplanar lines
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if two parallel planes are cut by a third plane, then the lines of intersection are parallel
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Theorem 3-1
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two nonadjacent interior angles on opposite sides of the transversal
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Alternate interior angles
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Same-Side Interior Angles
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two interior angles on the same side of the transversal
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two angles in corresponding positions relative to the two lines
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Corresponding angles
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if two parallel lines are cut by a transversal, then corresponding angles are congruent
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Postulate 10
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if two parallel lines are cut by a transversal, then alternate interior angles are congruent
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Theorem 3-2
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if two parallel lines are cut by a transversal, then same-side interior angles are supplementary
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Theorem 3-3
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if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other ones also
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Theorem 3-4
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if two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel
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Postulate 11
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if two lines are cut by a transversal and alternate interior angles are congruent, then the liens are parallel
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Theorem 3-5
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if two lines are cut by a transversal and same-side interior angles are supplementary, then the lines are parallel
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Theorem 3-6
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in a plane two lines perpendicular to the same lines are parallel
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Theorem 3-7
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through a point outside a line, there is exactly one line parallel to the given line
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Theorem 3-8
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through a point outside a line, there is exactly one line perpendicular to the given line
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Theorem 3-9
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two lines parallel to a third line are parallel to each other
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Theorem 3-10
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Ways to prove two lines parallel
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show that a pair of corresponding angles are congruent
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Ways to prove two lines parallel
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show that a pair of alternate interior angles are congruent
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Ways to prove two lines parallel
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show that a pair of same-side interior angles are supplementary
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Ways to prove two lines parallel
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in a plane show that both lines are perpendicular to a third line
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Ways to prove two lines parallel
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show that both lines are parallel to a third line
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Triangle
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figure formed by three segments joining three noncollinear points
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Scalene Triangle
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no sides congruent
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Isosceles Triangle
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at least two sides congruent
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Equilateral triangle
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all sides congruent
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Acute triangle
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three acute angles
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Obtuse triangle
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one obtuse angle
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Right triangle
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one right angle
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Equiangular triangle
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all angles congruent
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the sum of the measures of the angles of a triangle is 180
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Theorem 3-11
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if two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent
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Corollary 3-1
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each angle of an equiangular triangle has measure of 60
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Corollary 3-2
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in a triangle, there can be at most one right angle or obtuse angle
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Corollary 3-3
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the acute angles of a right triangle are complementary
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Corollary 3-4
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the measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles
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Theorem 3-12
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Polygon
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many angles
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Convex Polygon
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polygon such that no line containing a side of the polygon contains a point in the interior of the polygon
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the sum of the measures of the angles of a convex polygon with n sides is (n-2)180
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Theorem 3-13
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the sum of the the measures of the exterior angles of any convex polygon, one angle at each vertex, is 360
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Theorem 3-14
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Inductive Reasoning
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reasoning that is widely used in science and everyday life
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Deductive Reasoning
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conclusion based on accepted statements (definitions, postulates, previous theorems, corollaries, and given information) conclusion MUST be true if hypotheses are true
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Inductive Reasoning
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Conclusion based on several past observations. conclusion is PROBABLY true, but not necessarily true
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Congruent
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having the same size and shape
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SSS postulate
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if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent
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SAS postulate
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if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent
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ASA postulate
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if two angles and the included sides of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent
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A way to prove two segments or two angles are congruent
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identify two triangles in which the two segments or angles are corresponding parts
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A way to prove two segments or two angles are congruent
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prove that the triangles are congruent
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A way to prove two segments or two angles are congruent
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state that the two parts are congruent, using the reason CPCTC
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Isosceles Triangle Theorem
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if two sides of a triangle are congruent, then the angles opposite those sides are congruent
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an equilateral triangle is also equiangular
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Corollary 4-1
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an equilateral triangle has three 60° angles
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Corollary 4-2
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the bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint
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Corollary 4-3
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if two angles of a triangle are congruent, then the sides opposite those angles are congruent
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Theorem 4-2
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an equiangular triangle is also equilateral
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Corollary 4-4
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AAS theorem
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if two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the triangls are congruent
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HL theorem
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if the hypotenuse and a leg of one right triangle
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Ways to prove triangles congruent
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SSS, SAS, ASA, AAS
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Ways to prove right triangles congruent
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HL
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Median
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segment from a vertex to the midpoint of the opposite side in a triangle
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Altitude
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perpendicular segment from a vertex to the line that contains the opposite side in a triangle
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Perpendicular bisector
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line (or ray or segment) that is perpendicular to the segment at its midpoint
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if a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment
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Theorem 4-5
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if a point is equidistant from the endpoints of a segment, then the point is on the perpendicular bisector of the segment
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Theorem 4-6
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if a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle
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Theorem 4-7
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if a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle
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Theorem 4-8
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opposite sides of a parallelogram are congruent
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Theorem 5-1
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opposite angles of a parallelogram are congruent
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Theorem 5-2
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diagonals of a parallelogram bisect each other
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Theorem 5-3
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if both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram
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Theorem 5-4
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if one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram
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Theorem 5-5
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if both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram
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Theorem 5-6
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if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram
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Theorem 5-7
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Ways to prove that a quadrilateral is a parallelogram
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show that BOTH pairs of opposite sides are parallel
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Ways to prove that a quadrilateral is a parallelogram
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show that BOTH pairs of opposite sides are congruent
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Ways to prove that a quadrilateral is a parallelogram
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show that ONE pair of opposite sides are both congruent and parallel
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Ways to prove that a quadrilateral is a parallelogram
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show that both pairs of opposite angles are congruent
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Ways to prove that a quadrilateral is a parallelogram
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show that the diagonals bisect each other
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if two lines are parallel, then all points on one line are equidistant from the other line
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Theorem 5-8
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if three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal
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Theorem 5-9
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a line that contains the midpoint of one side of a triangle and is parallel to another side passes through the midpoint of the third side
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Theorem 5-10
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the segment that joins the midpoints of two sides of a triangle is parallel to the third side and is half as long as the third side
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Theorem 5-11
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Rectangle
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quadrilateral with four right angles
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Rhombus
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quadrilateral with four congruent sides
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Square
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quadrilateral with four right angles and four congruent sides
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the diagonals of a rectangle are congruent
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Theorem 5-12
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the diagonals of a rhombus are perpendicular
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Theorem 5-13
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each diagonal of a rhombus bisects two angles of the rhombus
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Theorem 5-14
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the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices
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Theorem 5-15
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if an angle of a parallelogram is a right angle, then the parallelogram is a rectangle
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Theorem 5-16
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if two consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus
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Theorem 5-17
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base angles of an isosceles trapezoid are congruent
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Theorem 5-18
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the median of a trapezoid is parallel to the bases and has a length equal to the average of the base lengths
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Theorem 5-19
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Exterior Angle Inequality Theorem
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the measure of an exterior angle of a triangle is greater than the measure of either remote interior angle
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if not p, then not q
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Inverse
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if not q, then not p
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Contrapositive
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if p, then q
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Given statement
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if q, then p
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Converse
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How to write an Indirect Proof
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Assume temporarily that the conclusion is not true
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How to write an Indirect Proof
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reason logically until you reach a contradiction of a known fact
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How to write an Indirect Proof
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point out that the temporary assumption must be false, and that the conclusion must then be true
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if one side of a triangle is longer than a second side, then the angle opposite the first side is larger than the angle opposite the second side
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Theorem 6-2
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if one angle of a triangle is larger than a second angle, then the side opposite the first angle is longer than the side opposite the second angle
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Theorem 6-3
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the perpendicular segment from a point to a line is the shortest segment from the point to the line
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Corollary 6-1
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the perpendicular segment from a point to a plane is the shortest segment form the point to the plane
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Corollary 6-2
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the sum of the lengths of any two sides of a triangle is greater than the length of the third side
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Triangle Inequality
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SAS Inequality Theorem
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if two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first trangle is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle
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SSS Inequality Theorem
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if two sides of one triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third side of the second, then the included angle of the first triangle is larger than the included angle of the second
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Ratio
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quotient when the first number is divided by the second
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Property of Proportions
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a/b = c/d is equivalent to ad = bc
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Property of Proportions
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a/b = c/d is equivalent to a/c = b/d
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Property of Proportions
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a/b = c/d is equivalent to b/a = d/c
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Property of Proportions
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a/b = c/d is equivalent to a+b/b = c+d/d
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Property of Proportions
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if a/b = c/d = e/f =..., then a + c +e +.../b + d + f + ... = a/b = ...
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AA ~
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if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar
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SAS Similarity Theorem
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if an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar
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SSS Similarity Theorem
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if the sides of two triangles are in proportion, then the triangles are similar
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Triangle Proportionality Theorem
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if a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally
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Triangle Angle-Bisector Theorem
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if a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the other two sides
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set of all points
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space
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points all in one line
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collinear points
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points all in one plane
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coplanar points
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set of points that are in both figures
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intersection
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two lines that intersect to form right angles
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Perpendicular lines
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equation stating that two ratios are equal
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proportion
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corresponding angles are congruent and corresponding sides are in proportion
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similar
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ratio of similar polygons
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scale factor
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Corollary 7-1
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if three parallel lines intersect two transversals, then they divide the transversals proportionally
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auxiliary line
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line (or ray or segment) added to a diagram to help in a proof
