JSUNIL TUTORIAL
Panjabi colony gali 01
The word 'The word 'geometry' is derived from the Greek words 'geo' meaning 'earth' and 'metron', meaning 'measuring'
Geometry originated when man felt the need to measure his land. Ancient Egyptians were perhaps the first people to study geometry. Later, the Babylonians studied in a systematic way
Ancient Indian mathematicians namely, Aryabhata, Mahaveera, Bharkara, Brahmagupta and others have made great contributions for the growth of mathematics
Point It is an exact location. It is a fine dot which has neither length nor breadth nor thickness but has position i.e., it has no magnitude. It is denoted by capital letters A, B, C, O etc.
Line segment The straight path joining two points A and B is called a line segmen
Ray A line segment which can be extended in only one direction is called a ray .
Line When a line segment is extended indefinitely in both directions it forms Line .
Collinear points If two or more points lie on the same line, then they are called collinear points.
Non-collinear points Points which do not lie on the same line are called as non-collinear points. Example: E, F, G, H, I
Intersecting lines Two lines having a common point are called intersecting lines. The common point is known as the point of intersection.
Concurrent lines If two or more lines intersect at the same point, then they are known as concurrent lines.
Plane A plane is a surface such that every point of the line joining any two points on it, lies on it. Example: Surface of a smooth wall, surface of a paper.
Statement A mathematical sentence which can be judged to be true or false is called a statement. Example: 5 + 3 = 8 is a statement.
Proof The course of reasoning, which establishes the truth or falsity of a statement is called proof.
Axioms The self-evident truths or the basic facts which are accepted without any proof are called axioms. Example: A line contains infinitely many points. Things which are equal to the same things are equal to each other.
Theorem A statement that requires a proof is called a theorem.
Corollary A statement whose truth can be easily deduced from a theorem is a corollary.
Proposition A statement of something to be done or considered is called proposition. You are familiar with statements such as “If two straight lines intersect each other then the vertically opposite angles are equal”. “The angles opposite to equal sides of a triangle are equal”. Proposition is a discussion and is complete in itself. A later proposition depends on the earlier one. In geometry there are many such statements and they are called propositions.
Theorems and Propositions
Propositions are of two kinds namely
Theorem and Problems
A theorem is a generalised statement, which can be proved logically. A theorem has two parts, a hypothesis, which states the given facts and a conclusion which states the property to be proved. The two statements given above are examples of theorems. Theorems are proved using undefined terms, definitions, postulates and occasionally some axioms from algebra. A theorem is a generalised statement because it is always true. For example the statement or the proposition “If two straight lines intersect, then the vertically opposite angles are equal” is true for any two straight lines intersecting at a point. Such a statement is called the general enunciation. In the theorem stated above, “two lines intersect” is the hypothesis and “vertically opposite angles are equal” is the conclusion. It is the conclusion part that is to be proved logically. To prove a theorem is to demonstrate that the statement follows logically from other accepted statements, undefined terms, definitions or previously proved theorems.
Converse of a theorem If two statements are such that the hypothesis of one is the conclusion of the other and vice-versa then either of the statement is said to be the converse of the other. Examples: Consider the statement of a theorem "If a transversal intersects two parallel lines, then pairs of corresponding angles are equal”. This theorem has two parts. If (hypothesis) and then (conclusion). Let us interchange the hypothesis and conclusion and write the statement. "If a transversal cuts two other straight lines such that a pair of corresponding angles are equal, then the straight lines are parallel". Such a statement with the hypothesis and conclusion interchanged is called the converse of a given theorem.
teps to be followed while providing a theorem logically:
Read the statement of the theorem carefully.
Identify the data and what is to be proved.
Draw a diagram for the given data.
Write the data and what is to be proved by using suitable symbols, applicable to the figure drawn.
Analyse the logical steps to be followed in proving the theorem.
Based on the analysis, if there is need for the construction, do it with the help of dotted lines and write it under the step 'Construction'.
Write the logical proof step by step by stating reasons for each step.
Postulates A statement whose validity is accepted without proof is called a postulate. In addition to point, line plane etc, it is also necessary to start with certain other basic statements that are accepted without proof. In geometry these are called postulates. A postulate, though itself is an unproved statement, can be cited as a reason to support a step in a proof. Postulates are just like axioms in arithmetic and algebra, that they are accepted without proof.
Some of the postulates we use often are: The line containing any two points in a plane lies wholly in that plane.
An angle has only one and only one bisector.
Through any point outside a line, one and only one perpendicular can be drawn to the given line.
A segment has one and only one mid point.
Linear pair postulate: If a ray stands on a line, then the sum of the two adjacent angles so formed is 180o.
Activity 1: Mark two distinct points A and B on the plane of your note book. Can you draw a line passing through A and B?
By experience we find that we can draw only one line through two distinct points A and B. Hence "Given any two distinct points in a plane, there exists one and only line containing them". This is a self-evident truth. Hence it is a postulate.
Activity 2: Draw m || l, draw n|| l. Measure the perpendicular distance between m and n at many points.
We will find this to be same at all points. This means by the property of parallel line m||n. From the above activity we can conclude that "Two lines which are parallel to the same line, are parallel to themselves". This is a postulate on parallel lines.
Angle When two straight lines meet at a point they form an angle.
The point at which the arms meet (O) is known as the vertex of the angle.
The amount of turning from one arm (OA) to other (OB ) is called the measure of the angle (ÐAOB) and written as m ÐAOB.
An angle is measured in degrees, minutes and seconds.
If a ray rotates about the starting initial position, in anticlockwise direction, comes back to its original position after 1 complete revolution then it has rotated through 360o.
1 complete rotation is divided into 360 equal parts. Each part is 1o.
Each part (1o) is divided into 60 equal parts, each part measures one minute, written as 1'. 1' is divided into 60 equal parts, each part measures 1 second, written as 1". Degrees -----> minutes --------> seconds 1o = 60'
Types of angles:
Right angle An angle whose measure is 90o is called a right angle.
Acute angle An angle whose measure is less then one right angle (i.e., less than 90o), is called an acute angle.
Obtuse angle An angle whose measure is more than one right angle and less than two right angles (i.e., less than 180o and more than 90o) is called an obtuse angle. 
Straight angle An angle whose measure is 180o is called a straight angle.
Straight angle An angle whose measure is 180o is called a straight angle. Reflex angle An angle whose measure is more than 180o and less than 360o is called a reflex angle.
Complete angle An angle whose measure is 360o is called a complete angle.
Equal angles Two angles are said to be equal, if they have the same measure.
Adjacent angles Two angles having a common vertex and a common arm, such that the other arms of these angles are on opposite sides of the common arm, are called adjacent angles.
Complementary angles If the sum of the two angles is one right angle (i.e., 90o), they are called complementary angles.
Supplementary angles Two angles are said to be supplementary, if the sum of their measures is 180o. Example: Angles measuring 130o and 50o are supplementary angles. Two supplementary angles are the supplement of each other.
Vertically opposite angles When two straight lines intersect each other at a point, the pairs of opposite angles so formed are called vertically opposite angles.
Angles Ð1 and Ð3 and angles Ð2 and Ð4 are vertically opposite angles. Note: Vertically opposite angles are always equal.
Bisector of an angle If a ray or a straight line passing through the vertex of that angle, divides the angle into two angles of equal measurement, then that line is known as the Bisector of that angle.
Linear pair of angles Two adjacent angles are said to form a linear pair of angles, if their non-common arms are two opposite rays.
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