Euclidean Geometry -
Definitions
Euclid - (Third Century B.C.)
Undefined terms
Before we begin our defintions, it must be stated that a few terms in geometry (three to be exact) are terms which have no defintion. The terms point, line, and plane are called "undefined" because they cannot be described without using words which are themselves undefined. These terms are fundamentally important in the study of geometry since we need these terms to describe more complex objects such as circles, quadrilaterals, and triangles.
Properties
We also must introduce some fundamental properties from algebra into our geometry. These properties are necessary in order to establish our some of the first theorems.
Commutative Property of Addition: a + b = b + a
Commutative Property of Multiplication: ab = ba
Associative Property of Addition: (a + b) + c = a + (b + c)
Associative Property of Multiplication: (ab)c = a(bc)
Distributive Property: a(b + c) = ab + ac
Addition Property of Equality: If a = b and c = d, then a + c = b + d.
Subtraction Property of Equality: If a = b and c = d, then a - c = b - d.
Multiplication Property of Equality: If a = b and c = d, then ac = bd.
Division Property of Equality: If a = b and c does not equal zero, then a/c = b/c.
Transitive Property of Equality: If a = b and b = c, then a = c.
Addition Property of Inequalities: If a < b and x < y, then a + x < b + y.
Multiplication Property of Inequalities: If x < y and a > 0, then ax < ay.
Transitive Property of Inequalities: If x < y and y < z, then x < z.
Reflexive Property: a = a, for every a.
Symmetric Property: If a = b, then b = a.
Trichotomy Property: For every x and y, one and only one of the following conditions holds:
x < y, x = y, x > y
Definitions
Definition of Union - The union of two sets is the set of all elements that belong to one or both sets.
Definition of Intersection - The intersection of two sets is the set of all elements common to the sets.
Existence of Square Roots - Every positive number has exactly one positive square root.
Definition of Distance Between Two Points - 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 PQ.
Definition of Coordinate - A correspondence of the sort described in the Ruler Postulate is called a coordinate system. The number corresponding to a given point is called the coordinate of the point.
Definition of Between - B is between A and C if A, B, and C are different points of the same line, and AB + BC = AC. When B is between A and C, we write A-B-C or C-B-A.
Definition of Segment - For any two points A and B, the segment 
is the union of A and B, and all points that are between A and B. The points A and B are called endpoints of .
Definition of Segment Length - The number AB is called the length of the segment .
Definition of Ray - Let A and B be points. The ray 
is the union of and the set of all points C for which A-B-C. The point A is called the endpoint of .
Definition of Opposite Rays - If A is between B and C, then and 
are opposite rays.
Definition of Midpoint - A point B is called a midpoint of 
if B is between A and C and AB = BC.
Definition of Bisect - The midpoint of a segment is said to bisect the segment. The midpoint of a segment or any line, plane, ray, or segment which contains the midpoint and does not contain is called a bisector of .
Definition of Space - Space is the set of all points.
Definition of Collinear - A set of points is collinear if there is a line which contains all the points of the set.
Definition of Coplanar - A set of points is coplanar if there is a plane which contains all points of the set.
Definition of Convex - A set M is called convex if for every two points P and Q of the set, the entire segment 
lies in M.
Definition of Half-Planes - 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 opposide sides of L.
Definition of Half-Spaces - The two sets described in the Space Seperation Postulate are called half-spaces, and the given plane is called the face of each of them.
Definition of Angle - If two rays have the same end point, but do not lie on the same line, then their union is called an angle. The two rays are called its sides, and their common endpoint is called its vertex. If the rays are and , then the angle is denoted by < BAC or < CAB.
Definition of Angle Interior - The interior of < BAC is the set of all points P in the plane of < BAC such that - P and B are on the same side of
, and - P and C are on the same side of
.
Definition of Angle Exterior - The exterior of of < BAC is the set of all points on the plane of < BAC that lie neither on the angle nor its interior.
Definition of Triangle - If A, B, and C are any three non collinear points, then the union of the segments , , and 
is called a triangle. The points A, B, and C are called its vertices, and the segments , , and are called its sides. Every triangle determines three angles, namely, < BAC, < ABC, and < ACB. The perimeter of the triangle is the sum of the lengths of its sides.
Definition of Triangle Interior - A point lies on the interior of a triangle if it lies in the interior of each of the angles of a triangle.
Definition of Triangle Exterior - A point lies in the exterior of a triangle if it lies in the plane of the triangle but does not lie on the triangle nor in the interior.
Definition of Angle Measure - The number given by the Angle Measurement Postulate is called the measure of < BAC and is written m< BAC.
Definition of a Linear Pair - If and 
are opposite rays, and is any other ray, then < BAC and < CAD form a linear pair.
Definition of Supplementary - If the sum of the measures of two angles is 180, then the angles are called supplementary, and each is called a supplement of the other.
Definition of a Right Angle - A right angle is an angle having measure 90.
Definition of an Acute Angle - An angle with measure less than 90 is called acute.
Definition of an Obtuse Angle - An angle with measure greater than 90 is called obtuse.
Definition of Complementary - If the sum of the measures of two angles is 90, then they are called complementary, and each of them is called a complement of the other.
Definition of Congruent Angles - Two angles with the same measure are called congruent.
Definition of Perpendicular - Two sets are perpendicular if- Each of them is a line, a ray, or a segment,
- They intersect, and
- The lines containing them are perpendicular.
Definition of Vertical Angles - Two angles are vertical angles if their sides form two pairs of opposite rays.
Definition of Congruent Segments - Segments are congruent if they have the same length.
Definition of Congruent Triangles - Given a correspondence ABC 
DEF between the vertices of two triangles. If every pair of corresponding angles are congruent, and every pair of corresponding angles are congruent, then the correspondence ABC DEF is called a congruence between the two triangles.
Definition of a Side of a Triangle - A side of a triangle is said to be included by the angles whose vertices are the end points of the segment.
Definition of an Angle of a Triangle - An angle of a triangle is said to be included by the sides of the triangle which lie in the sides of the angle.
Definition of Angle Bisector - If D is in the interior of < BAC, and < BAD 
< DAC, then bisects < BAD, and is called the bisector of < BAC.
Definition of an Isosceles Triangle - A triangle with two congruent sides is called isosceles. The remaining side is the base. The two angles that include the base are base angles. The angle opposite the base is the vertex angle.
Definition of an Equilateral Triangle - A triangle whose three sides are congruent is called equilateral.
Definition of a Scalene Triangle - A triangle, no two of whose sides are congruent, is called scalene.
Definition of an Equiangular Triangle - A triangle is equiangular if all three of its angles are congruent.
Definition of Median of a Triangle - A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side.
Definition of Angle Bisector of a Triangle - A segment is an angle bisector of a triangle if it lies in the ray which bisects an angle of a triangle, and its endpoints are the vertex of this angle and a point of the opposite side.
Definition of Perpendicular Bisector - In a given plane, the perpendicular bisector of a segment is the line which is perpendicular to the segment at its midpoint.
Definition of a Right Triangle - A right triangle is a triangle one of whose angles is a right angle. The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs.
Definition of Exterior Angle - If C is between A and D, then < BCD is an exterior angle of triangle ABC.
Definition of Remote Interior Angles - < A and < B of triangle ABC are called the remote interior angles of the exterior angles < BCD and < ACE.
Definition of Distance between a Line and an External Point - The distance 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.
Definition of Altitude - An altitude of a triangle is a perpendicular segment from a vertex of the triangle to the line containing the opposite side.
Definition of Skew - Two lines are skew if they do not lie in the same plane.
Definition of Parallel - Two lines are parallel if they are coplanar and they do not intersect.
Definition of Transversal - A transversal of two coplanar lines is a line which intersects them in two different places.
Definition of Alternate Interior Angles - Given two lines, L1 and L2, cut by a transversal T at points P and Q. Let A be a point on L1 and let B be a point on L2, such that A and B lie on opposite sides of T. Then < APQ and < PBQ are alternate interior angles.
Definition of Corresponding Angles - Given two lines cut by a transversal. If < x and < y are alternate interior angles, and < y and < z are vertical angles, then < x and < z are corresponding angles.
Definition of Interior Angles on the same side of the Transversal - Given two lines cut by a transversal. If < x and < y are alternate interior angles, < v and < w are alternate interior angles, and < v and < x form a linear pair, then < x and < w are interior angles on the same side of the transversal.
Definition of a Quadrilateral - Let A, B, C, and D be four points of the same plane. If no three of these points are collinear, and the segments , , 
, and 
intersect only at their end points, then the union of these four segments is called a quadrilateral. The four segments are called its sides, and the points A, B, C, and D are called its vertices. The angles < ABC, < BCD, < CDA, and < DAB are called its angles.
Definition of a Convex Quadrilateral - A quadrilateral is convex if no two of its vertices lie on opposite sides of a line containing a side of the quadrilateral.
Definition of the Parts of a Quadrilateral - Two sides of a quadrilateral are opposite if they do not intersect. Two of its angles are opposite if the do not have a side of the quadrilateral in common. Two sides are consecutive if they have a common end point. Two angles are consecutive if they have a side of the quadrilateral in common. A diagonal of a quadrilateral is a segment joining two nonconsecutive vertices.
Definition of a Parallelogram - A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
Definition of a Trapezoid - A trapezoid is a quadrilateral in which one and only one pair of opposite sides are parallel. The parallel sides are called the bases of the trapezoid. The segment joining the midpoints of the nonparallel sides is called the median.
Definition of Distance between Two Parallel Lines - The distance between two parallel lines is the distance from any point of one to the other.
Definition of a Rhombus - A rhombusis a parallelogram all of whose sides are congruent.
Definition of a Rectangle - A rectangle is a parallelogram all of whose angles are right angles.
Definition of a Square - A square is a rectangle all of whose sides are congruent.
Definition of Concurrent - Two or more lines are concurrent if there is a single point which lies on all of them. The common point is called the point of concurrency.
Definition of Triangular Region - A triangular region is the union of the triangle and its interior.
Definition of Polygonal Region - A polygonal region is the union of a finite number of triangular regions, in a plane, such that if two of these intersect, their intersection is either a point or a segment.
Definition of Area - The area of a polygonal region is the number assigned to it by the Area Postulate. The area of region R is denoted aR. This is pronounced area of R.
Definition of Altitude of a Trapezoid - The altitude of a trapezoid is the distance between the lines containing its parallel sides.
Definition of Proportion - Given two sequences A, B, C, ... and P, Q, R, ... of positive numbers. If A divided by P, is equal to B divided by Q, is equal to C divided by R, and so on, then the sequences A, B, C, ... and P, Q, R, ... are called proportional.
Definition of Geometric Mean - If A, B, C, are positive number, and A divided B equals B divided by C, then B is called the geometric mean between A and C.
Definition of Similar - Given a correspondence between two triangles. If corresponding angles are congruent, and corresponding sides are proportional, then the correspondence is called a similarity, and the triangles are said to be similar.