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Geometry postulates theorems and definitions

Geometry Postulates Theorems And Definitions

by timothyyoung.ty9, Sep. 2009

Subjects: geometry honors perryman postulates theorems

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189 Cards in this Set

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	The Distance Postulate	To every pair of different points there corresponds a unique positive number.
	The Ruler Postulate	The points of a line can be placed in correspondence with the real numbers in such a way that 1. to every point of the line there corresponds exactly one number 2. to every real number there corresponds exactly one point of the line and 3. the distance between any two points is the absolute value of the difference of the corresponding numbers
	The Ruler Placement Postulate	Given two points P and Q of a line, the coordinate system can be chosen in such a way that the coordinate of P is zero and the coordinate of Q is positive.
	The Line Postulate	For every two points there is exactly one line that contains both points.
	The Plane-Space Postulate	a. Every plane contains at least three noncollinear points. b. Space contains at least four noncoplanar points.
	The Flat Plane Postulate	If two points of a line lie in a plane, then the line lies in the same plane.
	The Plane Postulate	Any three points lie in at least one plane, and any three noncollinear points lie in exactly one point.
	Intersection of Planes Postulate	Given a line and a plane containing it. The points of the plane that do not lie on the line form two sets such that 1. each of the sets is convex, and 2. if P is in one of the sets and Q is in the other, then the segment PQ intersects the line.
	2-1 If a - b > 0, then...	a > b.
	2-2 If a = b + c and c > 0, then...	a > b.
	2-3 Let A, B, and C be points of a line, with coordinates x, y, and z respectively. If x < y< z, then...	A-B-C.
	2-4 If A, B, and C are three different points of the same line, then...	exactly one of them is between the other two.
	2-5 The Point-Plotting theorem. Let AB be a ray, and let x be a positive number. Then...	there is exactly one point P of AB such that AP = x.
	2-6 The Mid-Point Theorem. Every segment has...	exactly one mid-point.
	3-1 If two different lines intersect...	their intersection contains only one point.
	3-2 If a line intersects a plane not containing it, then...	the intersection contains only one point.
	3-3 Given a line and a point not on the line...	there is exactly one plane containing both.
	3-4 Given two intersecting lines...	there is exactly one plane containing both.
	The union...	of two sets is the set of all elements that belong to one or both sets.
	The intersection...	of two or more sets is the set of all elements common to the sets.
	Two sets intersect...	if there are one or more elements that are common to the sets.
	The distance...	between two points is the number given by the Distance Postulate. If the points are P and Q, then the distance is denoted by PQ.
	A coordinate system...	is a correspondence of the sort described in the Ruler Postulate.
	The coordinate of the point...	is the number corresponding to a given point.
	B is between A and C if...	1. A, B, and C are different points of the same line, and 2. AB + BC = AC. When B is between A and C, we write A-B-C or C-B-A.
	For any two points A and B, the segment AB is the...	union of A and B, and all points that are between A and B.
	The end points of AB are...	A and B.
	The number AB is called the...	length of the segment AB
	Let A and B be points. The ray AB is the union of...	1. AB and 2. the set of all points C for which A-B-C. The point A is called the end point of AB.
	If A is between B and C, then AB and AC are called...	opposite rays.
	A point B is called a midpoint of a segment AC if...	B is between A and C and AB = BC.
	The midpoint of a segment is to...	bisect the segment.
	The midpoint of a segment AB, or any line, plane, ray, or segment which contains the midpoint and does not contain AB, is called a...	bisector of AB.
	Space...	is the set of all points.
	A set of points is collinear if...	there is a line which contains all the points of the set.
	A set of points is coplanar if...	there is a plane which contains all points of the set.
	A set M is called convex if...	for every two points P and Q of the set, the entire segment PQ lies in M.
	The Plane Separation Postulate	Given a line and a plane containing it. The points of the plane that do not lie on the line form two sets such that (1) each of the sets is convex, and (2) if P is in one of the sets and Q is in the other, then the segment PQ intersects the line.
	Given a line L and a plane E containing it, the two sets described in the Plane Separation Postulate are called...	half-planes or sides of L, and L is called the edge of each of them
	If P lies in one of the half-planes and Q lies in the other, then we say that P and Q...	lie on opposite sides of L.
	The Space Separation Postulate	The points of space that do not lie in a given plane form two sets, such that (1) each of the sets is convex, and (2) if P is in one of the sets and Q is in the other, then the segment PQ intersects the plane.
	The two sets described in the Space Separation Postulate are called...	half-spaces, and the given plane is called the face of each of them.
	If two rays have the same end point, but do not lie on the same line, then their union is an...	angle.
	The two rays in an angle are called ___ and their common end point is called its ___.	sides, vertex
	The interior of angle BAC is the set of all points P in the plane of angle BAC such that...	(1) P and B are on the same side of line AC, and (2) P and C are on the same side of line AB.
	The exterior of angle BAC is...	the set of all points of the plane of angle BAC that lie neither on the angle nor in its interior.
	If A,B, and C are any three noncollinear points, then the union of the segments Ab, Ac, and BC is called a..., and is denoted by a triangle ABC.	triangle
	The points A,B, and C, of a triangle are called...	vertices.
	The segments AB, AC, and BC are called its...	sides.
	Every triangle ABC determines three angles, namely, angle BAC, ABC, and ACB. these are called the...	angles of triangle ABC.
	The ___ of a triangle is the sum of the lengths of its sides.	perimeter
	A point lies in the ___ of a triangle if it lies in the interior of each of the angles of the triangle.	interior
	A point lies in the ___ of a triangle if it lies in the plane of the triangle but does not lie on the triangle or in the interior.	exterior
	Degree	is the unit of measure.
	Measure	is the number of degrees in an angle/the number given by the Angle Measurement Postulate (written as m<BAC)
	The Angle Measurement Postulate	To every angle BAC there corresponds a real number between 0 and 180.
	The Angle Construction Postulate	Let ray AB be a ray on the edge of the half-plane H. For every number r between 0 and 180 there is exactly one ray AP, with P in H, such that measure PAB = r.
	The Angle Addition Postulate	If D is in the interior of angle BAC, then m<BAC = m<BAD + m<DAC.
	If ray AB and AD are opposite rays, and AC is any other ray, then angle BAC and angle CAD form a ___	linear pair.
	If the sum of the measures of two angle is 180, then the angles are called ___, and each is called a ___ of the other.	supplementary, supplement
	The Supplement Postulate	If two angles form a linear pair, then they are supplementary.
	A right angle	is an angle having measure 90.
	Acute	An angle with measure less than 90.
	Obtuse	An angle with measure greater than 90.
	If the sum of the measures of two angles is 90, then they are called ___, and each of them is called a ___ of the other.	complementary, complement.
	Congruent	Two angles with the same measure.
	Two rays are ___ if they are the sides of a right angle. Two lines are ___ if they contain a pair of perpendicular rays.	perpendicular, perpendicular
	Two sets are ___ if (1) each of them is a line, a ray, or a segment, (2) they intersect, and (3) the lines containing them are perpendicular.	perpendicular
	Reflexive Property	a = a, for every a.
	Symmetric Property	If a = b, then b = a.
	Transitive Property	If a = b and b = c, then a = c.
	4-1 Congruence between angles is an...	equivalence relation.
	4-2 If the angles in a linear pair are congruent,	then each of them is a right angle.
	4-3 If two angles are complementary,	then both are acute.
	4-4 Any two right angles are...	congruent.
	4-5 If two angles are both congruent and supplementary,	then each is a right angle.
	4-6 The Supplement Theorem. Supplements of congruent angles are...	congruent.
	4-7 The Complement Theorem. Complements of congruent angles are...	congruent.
	Two angles are ___ ___ if their sides form two pairs of opposite rays.	vertical angles
	4-8 The Vertical angle theorem. Vertical angles are...	congruent.
	4-9 If two lines are perpendicular,	they form four right angles.
	Angles are ___ if they have the same measure.	congruent
	Segments are ___ if they have the same length.	congruent
	If every pair of corresponding sides are congruent, and every pair of corresponding angles are congruent, then the correspondence ABC <-> DEF is called a...	congruence between the two triangles.
	A ___ of a triangle is said to be ___ by the angles whose vertices are the end points of the segment.	side, included
	An ___ of a triangle is said to be ___ by the sides of the triangle which lie in the sides of the angle.	angle, included
	5-1 Congruence for segments is...	an equivalence relation.
	5-2 Congruence for triangles is...	an equivalence relation
	The SAS Postulate	Every SAS correspondence is a congruence.
	The ASA Postulate	Every ASA correspondence is a congruence.
	The SSS Postulate	Every SSS correspondence is a congruence.
	If D is in the interior of <BAC, and <BAD is congruent to <DAC, then ray AD bisects <BAC, and ray AD is called the ___ of <BAC.	bisector
	5-3 The Angle Bisector Theorem	Every angle has one and only one bisector.
	5-4 The Isosceles Triangle Theorem	If two sides of a triangle are congruent, then the angles opposite these sides are congruent.
	A triangle with two congruence sides is called... The remaining side is the... The two angles that include the base are... The angles opposite the base is the...	isosceles, base, base angles, vertex angle.
	A triangle whose three sides are congruent is called...	equilateral.
	A triangle no two of whose sides are congruent is called...	scalene.
	A triangle is ___ if all three of its angles are congruent.	equilangular
	Corollary 5-4.1 Every equilateral triangle is...	equiangular
	5-5 If two angles of a triangle are congruent, then...	the sides opposite them are congruent.
	Corollary 5-5.1 Every equiangular triangle is...	equilateral.
	Let A, B, C, and D be four coplanar points. If no three of these points are collinear, and the segments AB, BC, CD, and DA intersect only at their ends points, then the union of the four segments is called a ___. The four segments are called its ___, and the points A, B, C, and D are called its ___. The angles DAB, ABC, BCD, and CDA are called its ___ and may be denoted briefly as <A, <B, <C, and <D. AC and BD are called its ___. The ___ of a quadrilateral is the sum of the lengths of its sides.	quadrilateral, sides, verticies, angles, diagnoals, perimeter
	If all four angles of a quadrilateral are right angles, then the quadrilateral is a ___.	rectangle
	If all four of the angles are right angles, and all four sides are congruent, then the quadrilateral is a ___.	square
	A ___ of a triangle is a segment whose end points are a vertex of the triangle and the midpoint of the opposite side.	median
	A segment is an ___ if (1) it lies in the ray which bisects an angle of the triangle, and (2) its end points are the vertex of this angle and a point of the opposite side.	angle bisector of a triangle
	6-1 In a given plane, through a given point of a given line...	there is one and only one line perpendicular to the given line.
	In a given plane, the perpendicular bisector of a segment...	is the line which is perpendicular to the segment at its midpoint.
	6-2 The perpendicular bisector of a segment, in a plane,	is the set of all points of the plane that are equidistant from the end points of the segment.
	Corollary 6-2.1 Given a segment AB and a line L in the same plane. If two points of L are each equidistant from A and B, then L...	is the perpendicular bisector of AB.
	6-3 Through a given external point there is at least...	one line perpendicular to a given line.
	6-4 Through a given external point there is at most...	one line perpendicular to a given line.
	Corollary 6-4.1 No triangle has...	two right angles.
	A right triangle is a traingle one of whose...	angles is a right angle.
	The side opposite the right angle is called the___, and the other two sides are called the ___.	hypotenuse, legs
	6-5 If M is between A and C on a line L, then...	M and A are on the same side of any other line that contains C.
	6-6 If M is between B and C, and A is any point not on line BC, then M...	is in the interior of angle BAC.
	7-5 If two sides of a triangle are of unequal length...	then the angles opposite them are of unequal measure, and the larger angle is opposite the longer side.
	One segment is less than (or shorter than) another...	if its length is less.
	One angle is less than (or smaller than) another...	if its measure is less.
	The Parts Theorem	The whole is greater than any one of its parts.
	If C is between A and d, then angle BCD is a ___ of triangle ABC.	exterior angle
	Angle A and B of triangle ABC are called the ___ of the exterior angles BCD and ACE.	remote interior angles
	The Exterior Angle Theorem	An exterior angle of a triangle is greater than each of its remote interior angles.
	Corollary 7-2.1 If a triangle has one right angle...	then its other angles are acute.
	The SAA Theorem	Every SAA correspondence is a congruence.
	The Hypotenuse-Leg Theorem	Given a correspondence between two right angles. If the hypotenuse and one leg of one of the triangles are congruent to the corresponding parts of the second triangle, then the correspondence is a congruence.
	7-6 If two angles of a triangle are not congruent...	then the sides opposite them are not congruent, and the longer side is opposite the larger angle.
	The First Minimum Theorem	The shortest segment joining a point to a line is the perpendicular segment.
	The ___ between a line and an external point is the length of the perpendicular segment from the point to the line. The distance between a line and a point on the line is defined to be zero.	distance
	The Triangle Inequality	The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
	The Hinge Theorem	Given triangle ABC and triangle DEF, with AB = DE and AC = DF. If angle A is greater than angle D, then line segment BC is greater than EF.
	The Converse Hinge Theorem	Give triangle ABC and triangle DEF, with AB = DE and AC = DF. If line segment BC is greater than EF, then angle A is greater than D.
	An ___ of a triangle is a perpendicular segment from a vertex of the triangle to the line containing the opposite side.	altitude
	Two lines are ___ if they do not lie in the same plane. Two lines are ___ if (1) they are coplanar and (2) they do not intersect.	skew lines, parallel
	9-1 Two parallel lines lie...	in exactly one plane.
	9-2 In a plane, if two lines are both perpendicular to the same line, then...	they are parallel
	9-3 Existence of Parallels. Let L be a line and let P be a point not on L. Then there is at least one line through	P, parallel to L.
	A transversal of two coplanar lines is...	a line which intersects them in two different points.
	Given two lines L1 and L2, cut by transversal T at points P and Q. Let A be a point of L1 and let B be a point of L2, such that A and B lie on opposite sides of T. Then angle APQ and PQB are ___.	alternate interior angles.
	If two lines are cut by a transversal, and one pair of alternate interior angles are congruent, then the other pair of alternate interior angles are also...	congruent.
	9-5 AIP Theorem Given two lines cut by a transversal. If a pair of alternate interior angles are congruent, then the lines are...	parallel.
	Given two lines cut by a transveral. If angle x and angle y are alternate interior angles, and angle y and angle z are vertical angles, then angle x and angle z are...	corresponding angles.
	Given two lines cut by a transversal. If (1) angle x and angle y are alternate interior angles, (2) angle v and angle w are alternate interior angles, and (3) angle v and angle x form a linear pair, then angle x and angle w are ___	interior angles on the same side of the transversal.
	9-6 Given two lines cut by a transversal. If a pair of corresponding angles are congruent, then a pair of alternate interior angles are...	congruent.
	9-7 The CAP Theorem Given two lines cut by a transversal. If a pair of corresponding angles are congruent, then...	the lines are parallel.
	9-8 Given two lines cut by a transversal. If a pair of interior angles on the same side of the transversal are supplementary...	the lines are parallel.
	The Parallel Postulate	Through a given external point there is only one parallel to a given line.
	9-9 The PAI Theorem If two parallel lines are cut by a transversal, then alternate interior angles are...	congruent.
	Corollary 9-9.1 The PCA Corollary If two parallel lines are cut by a transversal, each pair of corresponding angles are...	congruent.
	Corollary 9-9.2 If two parallel lines are cut by a transversal, the interior angles on the same side of the transversal are...	supplementary.
	9-10 In a plane, if a line intersects one of two parallel lines in only one point, then...	it intersects the other.
	9-11 In a plane, if two lines are each parallel to a third line, then...	they are parallel to each other.
	9-12 In a plane, if a line is perpendicular to one of two parallel lines it is...	perpendicular to the other.
	9-13 For every triangle, the sum of the measures of the angles is...	180.
	Corollary 9-13.1 Given a correspondence between two triangles. If two pairs of corresponding angles are congruent, then the third pair of corresponding angles are also	congruent.
	Corollary 9-13.2 The acute angles of a right triangle are...	complementary.
	Corollary 9-13.3 For any triangle, the measure of an exterior angle is the...	sum of the measures of two remote interior angles.
	Let A, B, C, and D be four points of the same plane. If no three of these points are collinear, and the segments AB, BC, CD, and DA intersect only at their end points, then the union of these four segments is called a ___. The four segments are called its ___, and the points A, B, C, and D are called its ___. The angles DAB, ABC, BCD and CDA are called its ___.	quadrilateral, sides, vertices, angles.
	A quadrilateral is ___ if no two of its vertices lie on opposite sides of a line containing a side of the quadrilateral.	convex
	Two sides of a quadrilateral are ___ if they do not intersect. Two of its angles are ___ if they do not have a side of the quadrilateral in common. Two sides are ___ if they have a common end point. Two angles are ___ if they have a side of the quadrilateral in common. A ___ of a quadrilateral is a segment joining two nonconsecutive vertices.	opposite, opposite, consecutive, consecutive, diagonal.
	A ___ is a quadrilateral in which both pairs of opposite sides are parallel.	parallelogram
	A ___ is a quadrilateral in which one and only one pair of opposite sides are parallel. The parallel sides are called the ___ of the trapezoid. The segment joining the midpoints of the nonparallel sides is called the ___.	trapezoid, bases, median
	9-14 Each diagonal separates a parallelogram into...	two congruent triangles.
	9-15 In a parallelogram, any two opposite sides are...	congruent
	Corollary 9-15.1 If two lines are parallel, then all points of each line are...	equidistant from the other line.
	The ___ is the distance from any point of one to the other.	distance between two parallel lines
	9-16 In a parallelogram, any two opposite angles are...	congruent
	9-17 In a parallelogram, any two consecutive angles are...	supplementary.
	9-18 The diagonals of a parallelogram...	bisect each other.
	9-19 Given a quadrilateral in which both pairs of opposite sides are congruent. Then the quadrilateral is a...	parallelogram.
	9-20 If two sides of a quadrilateral are parallel and congruent, then the quadrilateral is a...	parallelogram.
	9-21 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a...	parallelogram.
	9-22 The Midline Theorem The segment between the midpoints of two sides of a triangle is parallel to the...	third side and half as long.
	A ___ is a parallelogram all of whose sides are congruent.	rhombus
	A ___ is a parallelogram all of whose angles are right angles.	rectangle
	A ___ is a rectangle all of whose sides are congruent.	square
	9-23 If a parallelogram has one right angle, then it has...	four right angles, and the parallelogram is a rectangle.
	9-24 In a rhombus, the diagonals are...	perpendicular to one another.
	9-25 If the diagonals of a quadrilateral bisect each other and are perpendicular, then the quadrilateral is a...	rhombus.
	9-26 The median to the hypotenuse of a right triangle is...	half as long as the hypotenuse.
	9-27 The 30-60-90 Triangle Theorem. If an acute angle of a right triangle has measure 30, then the opposite side is...	half as long as the hypotenuse.
	9-28 If one leg of a right triangle is half as long as the hypotenuse, then the opposite angle has measure...	30.
	If a transversal intersects two lines L1, L2 in points A and B, then we say that L1 and L2 ___ the segment AB on the transversal.	intercept
	9-29 If three parallel lines intercept congruent segments on one transversal T, then they intercept congruent segments on every transversal...	T1 which is parallel to T.
	9-30 If three parallel lines intercept congruent segments on one transversal, then they intercept...	congruent segments on any other transversal.
	Corollary 9-30.1 If three or more parallel lines intercept congruent segments on one transversal, then they intercept...	congruent segments on any other transversal.
	Two or more lines are ___ if there is a single point which lies on all of them. The common point is called the ___.	concurrent, point of concurrency
	9-31 The Median Concurrence Theorem. the medians of every triangle are concurrent. Their point of concurrency is...	two-thirds of the way along each median, from the vertex to the opposite side.

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