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Chapter 8: Triangles

    Introduction

    Similar Figures

    Similarity of Triangles

    Criteria for Similarity of Triangles

    Pythagoras’ Theorem

    Different Forms Of Theoretical Statements

    Summary

    Enhanced Curriculum Support

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Chapter 8: Triangles > Different Forms Of Theoretical Statements

Different Forms Of Theoretical Statements

1. Negation of a statement :

We have a statement and if we add “Not” after the statement, we will get a new statement; which is called negation of the statement.

For example, take a statement “△ABC is a equilateral”. If we denote it by “p”, we can write like this.

p : Triangle ABC is equilateral and its negation will be “Triangle ABC is not equilateral”. Negation of statement p is denoted by ~p; and read as negation of p. The statement ~p negates the assertion that the statement p makes.

When we write the negation of the statements we would be careful that there should no confusion; in understanding the statement.

Observe this example carefully

P : All irrational numbers are real numbers. We can write negation of p like this.

i) ~p : All irrational numbers are not real numbers. How do we decide this negation is true or false? We use the following criterion “Let p be a statement and ~p its negation. Then ~p is false whenever p is true and ~p is true whenever p is false.

For example        s : 2 + 2 = 4 is True

       ~s : 2 + 2 ≠ 4 is False

2. Converse of a statement :

A sentence which is either true or false is called a simple statement. If we combine two simple statements then we will get a compound statement. Connecting two simple statements with the use of the words “If and then” will give a compound statement which is called implication (or) conditional.

Combining two simple statements p & q using if and then, we get p implies q which can be denoted by p ⇒ q. In this p ⇒ q, suppose we interchange p and q, we get q ⇒ p. This is called its converse.

Example : p ⇒ q : In △ABC, if AB = AC then ∠C = ∠B

Converse q ⇒ p : In △ABC, if ∠C = ∠B then AB = AC

3. Proof by contradiction :

In this proof by contradiction, we assume the negation of the statement as true; which we have to prove. In the process of proving we get contradiction somewhere. Then, we realize that this contradiction occurs because of our wrong assumption which is negation is true. Therefore we conclude that the original statement is true.