Geometry Postulates And Theorems

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Introduction to Geometry Postulate and Theorems:

Geometry:

Geometry is branch of mathematics, which is explained shapes or sizes of mathematical objects and their postulates. Geometry is main element to prove the properties, postulates and theorems.

Postulate or Axiom:

Postulate is a basic principle or fundamental theorem or general proof of any subject. In geometry, it is used for solving the proofs and understands the truth of related topic logically.

An postulate or axiom is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision. Axioms cannot be derived by principles of deduction otherwise they would be classified as theorems.

Types of axiom:

Axiom is classified into two types:

1) Logical axioms:

Logical axioms are generally statements that are obtained to be commonly factual.

2) Non – logical axioms:

It is defined as the properties of domain for particular mathematical theories. Usually, a non-logical axiom is not a self-evident fact.

Theorem:

Generally, theorem ...
... is defined as a statement used for finding some basic property and requirement of its proof in geometry. Thus theorems are named also as properties, rules and statements. To prove a theorem, we have to find the earlier found properties.

Theorem is one of the statement, which is proved from already exists statements. Theorem contains two elements, called as hypotheses and conclusions.

Postulates in Geometry :

The geometry postulates used in line segment are followed by,

Line: A line can illustrate between two points only.
Parallel lines: These are straight lines in the similar plane and do not meet together. They may extend in any direction
Intersection: The intersection of two lines meet single point called as intersection point.
Midpoint: A line section contains single midpoint only.
Every line and every plane are locations of points.
All lines include a coordinate structure.
Any straight-line section can be enlarged indefinitely in a straight line.
The geometry postulates used in angles are followed by,

Angle: It is measure of direction, which has two rays dividing by general end.
The angle of two straight lines, which is meeting, is inclination to each other.
Vertex: The assembling point of two lines is called the vertex.
Vertex angle: An angle is reverse to the base.
Right angle: Every right angle is congruent angles. A right angle is greater than an acute angle and less than obtuse angle.
Complementary angle: An angle is equal to single right angle.
Supplementary angle: An angle is equal to two right angles.
Bisector: It is a ray of interior angle, which bisects that angle. An angle contains single bisector only.
If two points recline in a flat surface, the line surrounding the points reclines in the flat surface.
The intersection of two planes meets single line.
The geometry postulates used in quadrilaterals are followed by,

Square: It is one of the quadrilateral which has equal sides with every angle is right angle.
Parallelogram: It is quadrilaterals with opposite sides are similar (parallel)
Quadrilateral: Four straight lines enclose a quadrilateral.
Circle: A circle is a plane surrounded by single line, called the circumference. A circle has angle 360 about their circumference.
Diameter: A straight line during the center of the circle is called diameter.
Radius: A straight line from the center of the circle is called radius of the circle.
Triangle: It is circumscribed with three straight lines.
Equilateral triangle: A triangle, which has three equal sides and interior angles, is called as an equilateral triangle or regular triangle.
Isosceles triangle: A triangle, which has two equal sides and interior angles, is called as an isosceles triangle.
Scalene triangle: A triangle, which has unequal sides and different interior angles, is called as scalene triangle or irregular triangle.
Right triangle: It is a triangle, which has single right angle.
Polygon: A polygon is bounded with over four straight lines.
Regular polygon: A polygon, which has identical sides and identical angles, is called regular polygon.
Triangle Congruence Postulate:

Side-Side-Side (SSS): If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
Side-Angle-Side (SAS): Side Angle Side theorem states that, If two sides and the included angle of one triangle are congruent to the corresponding part of another triangle, the triangles are congruent.
Angle-Side-Angle (ASA): Angle Side Angle theorem states that, If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
Equality Postulates:

a) Equality of addition:

Let assume l, m, n are real numbers. If l =m, then it can be written as l+n = m+n.

b) Equality of subtraction:

Let assume l, m, n are real numbers. If l =m, then it can be written as l-n = m-n.

c) Equality of multiplication:

Let assume l, m, n are real numbers. If l =m, then it can be written as l*n = m*n.

d) Equality of division:

Let assume l, n, m are real numbers (n =/ 0). If l =m, then it can be written as l/n = m/n.

e) Reflexive property:

Let assume ‘a’ is a real number, and then it reflects by itself. That the real number equals itself as, a = a.

f) Symmetric property:

Let assume a and b are real numbers. If a = b, then it can be written as, a =b. The order of equality is not considered.

g) Transitive property:

Let assume a, b, and c are real numbers. If a = b and b = c, then it can be written as, c =a. Thus, the two quantities identical to the same quantity are identical to each other

h) Distributive property:

Let assume p, q, r are real numbers. Then it states that as follows,

p(q+r) = pq+pr

Theorems in Geometry :

The basic geometry theorems are,

Line Intersection Theorem: Two different lines intersect in at most one point.
Betweenness Theorem: If C is between A and B and on AB, then AC + CB = AB.
Related Theorem: If A, B, and C are distinct points and AC + CB = AB, then C lies on AB.
Pythagorean Theorem: a2 + b2 = c2, if c is the hypotenuse.
The geometry theorems used in triangles are followed by,

The straight line that bisects the vertex angle of an isosceles triangle is the perpendicular bisector of the base.
In a right triangle the square drawn on the side opposite the right angle will equal the squares drawn on the sides that make the right angle.
If a transversal line intersect two parallel line then the Alternate Exterior angles are equal.
If a transversal line intersect two parallel line then the Alternate interior angles are equal.

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