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Lines and Angles

The Axioms of Euclidean Geometry (1) A straight line may be draw between any two points.

(2) Any terminated straight line may be extended indefinitely

(3) A circle may be drawn with any given point as center and any given radius.

(4) All right angles are equal.

(5) If α + β < 180 ° then L1 and L2 will meet at some point to the left of L3 or at the side of angles.

Fundamental Concepts

Axiom: The basic facts which are taken for granted without proof are called axioms.
Ex: (i) things which are equal to the same thing are equal to one another;
(ii) Halves of equals are equal;
(iii) A line contains infinitely many points;
(iv) The whole is greater than each of its parts.

The following axioms, corresponding to the first four rules of arithmetic, are among those most commonly used in geometry:
(i) Addition: If equals are added to equals, the sums are equal;
(ii) Subtraction: If equals are taken from equals, the remainders are equal;
(iii) Multiplication: Things which are the same multiples of equals are equal to one another;
(iv) Division: Things which are the same parts of equals are equal to one another.

Statement: A sentence which can be judged to be true or false is called a statement.
Ex: (i) The sum of the angles of a quadrilateral is 360°, is a true statement;
(ii) The sum of the angles of a triangle is 360°, is a false statement;
(iii) a + 2 < 10 is a sentence but not a statement.

Proof of a statement: Providing a geometrical result by using logical reasonings on previously proved and known results is called a proof of a statement.

Theorem: A statement that requires a proof is called a theorem. Establishing the truth of a theorem is known as proving the theorem.
Ex: (i) The sum of all the angles of a quadrilateral is 360°;
(ii) If one side of a triangle is produced, then the exterior angle so formed is equal to the sum of the interior opposite angles.

Corollary: A statement whose truth can easily be deduced from a theorem is called its corollary.
Ex: (i) If two angles of one triangle are equal to two angles of another triangle, then the third angles of both the triangles are also equal. This is the corollary for the theorem: the sum of all the angles of a triangle is 180°;
(ii) An exterior angle of a triangle is greater than either of the two interior opposite angles. This is the corollary for the theorem: if one side of a triangle is produced, then the exterior angle so formed is equal to the sum of the interior opposite angles.

Points and Lines
Point and Line

Point: A point is an exact location or position on a plane surface. The points are dimensionless. That is, a point has no width, height or length. We usually represent a point with a dot on paper, but small as such a dot may be, it still has some length and breadth, and is therefore not actually a geometrical point. The smaller the dot however, the more nearly it represents a point. Points are denoted by capital letters like A, B, X, Y, etc. In the adjacent figure, there are six points A, B, C, D, P and X.

Line: A line is a set of points that extends endlessly in both directions, that is, a line has no end points. A line is one-dimensional, having length, but no width or height. Lines are uniquely determined by two points. Thus, we denote the name of a line passing through the points P and Q as , where the two-headed arrow signifies that the line passes through those unique points and extends endlessly in both directions. In the adjacent figure, there are two lines and . Sometimes lowercase letters like 'l', 'm', 'n' are used to denote a line.

Line segment, Ray and Collinear points
Line Segment, Ray and Collinear Points

Line segment: A line segment is the part of a line consisting of two endpoints and all points between them. It has a definite length. We write the name of a line segment with endpoints A and B as AB. The distance between two points A and B is equal to the length of the line segment AB. An adjacent figure shows the line segment AB.

Ray: A ray is the part of a line that begins at a certain point and extends indefinitely in one direction. It has one endpoint, which marks the position from where it begins. A ray beginning at the point P that passes through the point Q is denoted as . This notation shows that the ray begins at point P and extends indefinitely in the direction of point Q. An adjacent figure shows the ray .

Collinear points: Three or more than three points are said to be collinear, if there is a line which contains them all. In the adjacent figure, P, Q, R are collinear points, while X, Y, Z are non-collinear points.

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