Lesson Introduction to basic postulates and Axioms in Geometry

The Lesson will deal with some common postulates in geometry which are widely used.

Postulates

In geometry there are some basic statements called postulates which are not required to be proved and are accepted as they are. These postulates serve as a basis for proving other statements and explain any undefined terms in geometry.

Some basic postulates in geometry accepted are:

Point,Line and Plane Postulates:

1.Unique Line Assumption: For every two points A, B there exists a line l that contains each of the points A, B.

2.For every two points A, B there exists no more than one line that contains each of the points A, B.

3.For any three points A, B, C that do not lie on the same line or are non-collinear there exists a plane P that contains each of the points A, B, C. Also for every plane there exists atleast 1 point which is contained in it.

4.If two points A, B of a line lie in a plane P then every point of the line will lie on the plane P.

5.There exists a unique distance between any two points in space and line is the shortest path between them.

6.If two different planes intersect each other or have a point in common then there intersection point is a line.

7.For any two points A and C in space, there will always exists at least one point B which lies on the line AC such that C lies between A and B.

8.For any line L and a point A in space which does not lie on the line. There exists at most one line in the plane that contains point A and does not intersect line L.

9.For any point A on the line L,exactly one line perpendicular to the first line L can be drawn at A.

10.For any point A in space not on the line L,exactly one line perpendicular to the first line L can be drawn which contains A.

Segment Addition Postulate :

For three points A,B and C which are collinear and B is between A and C, then AB + BC = AC

Angle Addition Postulate :

If point B is in the interior of an angle AOC, then

Angle AOB + Angle BOC = Angle AOC

If angle AOC is a straight angle,or AOC is a straight line, then

Angle AOB + Angle BOC = 180 degrees

Euclid's Postulates

1. A line segment is defined by two points.

2. A line segment is a part of line and can be extended indefinitely along the line.

3. All right angles are congruent to each other.

4. For any two lines cut by a transversal, if the sum of interior angles is less than 180 degrees then the angles will lie on same side of transversal. This sum is maximum and equal to 180 degrees when the lines are parallel.

Postulates of Congruency:

1.If two line segments A'B' and A"B" are congruent to the same line segment AB, then segments A'B' and A"B" are also congruent to each other.

2.SAS congruency : If for two triangles ABC and A'B'C', AB is congruent to A'B', AC congruent to A'C' and angle (BAC) is congruent to angle (B'A'C') is true, then by the condition of congruency angle (ABC) is also congruent to angle (A'B'C'). Similarly all conditions of congruency (SSS and ASA) also holds true.

Postulates on Polygon Inequality

1.Postulate on Triangle Inequality : The sum of the lengths of any two sides of any triangle taken at a time is always greater than the length of the third side.

2.Postulate on Quadrilateral Inequality : The sum of the lengths of any 3 sides of a quadrilateral taken at a time is always greater than the length of the fourth side.

For further reading on postulates refer to Wikipedia