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| Important terms |
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| Statement |
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| A mathematical sentence which can be judged to be true or false is called a statement. |
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| Example: |
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| 5 + 3 = 8 is a statement. |
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| Proof |
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| The course of reasoning, which establishes the truth or falsity of a statement is called proof. |
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| Axioms |
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| The self-evident truths or the basic facts which are accepted without any proof are called axioms. |
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| Example: |
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 A line contains infinitely many points. |
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 Things which are equal to the same things are equal to each other. |
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| Theorem |
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| A statement that requires a proof is called a theorem. |
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| Corollary |
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| A statement whose truth can be easily deduced from a theorem is a corollary. |
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| Proposition |
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| A statement of something to be done or considered is called proposition. |
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| You are familiar with statements such as |
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| “If two straight lines intersect each other then the vertically opposite angles are equal”. |
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| “The angles opposite to equal sides of a triangle are equal”. |
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| Proposition is a discussion and is complete in itself. A later proposition depends on the earlier one. |
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| In geometry there are many such statements and they are called propositions. |
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| Theorems and Propositions |
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| Propositions are of two kinds namely |
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 Theorem and |
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 Problems |
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| 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. |
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| Theorems are proved using undefined terms, definitions, postulates and occasionally some axioms from algebra. |
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| 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. |
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| 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. |
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| Converse of a theorem |
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| 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. |
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| Examples: |
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| Consider the statement of a theorem |
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| "If a transversal intersects two parallel lines, then pairs of corresponding angles are equal”. |
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| This theorem has two parts. If (hypothesis) and then (conclusion). |
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| Let us interchange the hypothesis and conclusion and write the statement. |
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| "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. |
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| Steps to be followed while providing a theorem logically: |
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 Read the statement of the theorem carefully. |
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 Identify the data and what is to be proved. |
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 Draw a diagram for the given data. |
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 Write the data and what is to be proved by using suitable symbols, applicable to the figure drawn. |
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 Analyse the logical steps to be followed in proving the theorem. |
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 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'. |
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 Write the logical proof step by step by stating reasons for each step. |
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| Postulates |
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| A statement whose validity is accepted without proof is called a postulate. |
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| 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. |
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| 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. |
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| Some of the postulates we use often are: |
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 The line containing any two points in a plane lies wholly in that plane. |
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 An angle has only one and only one bisector. |
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 Through any point outside a line, one and only one perpendicular can be drawn to the given line. |
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 A segment has one and only one mid point. |
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 Linear pair postulate: If a ray stands on a line, then the sum of the two adjacent angles so formed is 180o. |
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| Activity 1: |
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| Mark two distinct points A and B on the plane of your note book. Can you draw a line passing through A and B? |
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| By experience we find that we can draw only one line through two distinct points A and B. |
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| Hence "Given any two distinct points in a plane, there exists one and only line containing them". |
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| This is a self-evident truth. Hence it is a postulate. |
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| Activity 2: |
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| Draw m || l, draw n|| l. |
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| Measure the perpendicular distance between m and n at many points. |
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| We will find this to be same at all points. This means by the property of parallel line m||n. |
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| From the above activity we can conclude that |
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| "Two lines which are parallel to the same line, are parallel to themselves". |
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| This is a postulate on parallel lines. |
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