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NCERT Solutions For Class 10 Maths Chapter 6 Triangles

For example, suppose a politician tells you, If you are interested in a clean government, then you should vote for me.

What he actually wants you to believe is that if you do not vote for him, then you may not ch 6 maths class 10 theorems kit a clean government. Similarly, if an advertisement tells you, The intelligent wear XYZ shoes, what the company wants you to conclude is that if you do not wear XYZ shoes, then you are not intelligent.

You can yourself observe that both the above statements may mislead the general public. So, if we understand the process of reasoning correctly, we do not fall into such traps unknowingly. The correct use of reasoning is at the core of mathematics, especially in constructing proofs.

In Class IX, you were introduced to the idea of proofs, and you actually proved many statements, especially in geometry.

Recall that a proof is made up of several mathematical statements, each of which is logically deduced from a previous statement in the proof, or from a theorem proved earlier, or an axiom, or the hypotheses.

The main tool, we use in constructing a proof, is the process of deductive reasoning. We start the study of this chapter with a review of what a mathematical statement is. Then, we proceed to sharpen our skills in deductive reasoning using several examples. We shall also deal with the concept of negation and finding the negation of a given statement. Then, ch 6 maths class 10 theorems kit discuss what it means to find the converse of a given statement.

Finally, we review the ingredients of a proof learnt in Class IX by analysing the proofs of several theorems. Here, we also discuss the idea of proof by contradiction, which you have come across in Class IX and many other chapters of this book.

Go and finish your homework is an order, not a statement. What a fantastic goal! Remember, in general, statements can be one of the following: always true always false ambiguous In Class IX, you have also studied that in mathematics, a statement is acceptable only if it is either always true or always false. So, ambiguous sentences are not considered as mathematical statements. Let us review our understanding with a few examples.

Example 1 : State whether the following statements are always true, always false or ambiguous. J ustify your answers. Solution : i This statement is always false, since astronomers have established that the Earth orbits the Sun. This depends on what the vehicle is vehicles can have 2, 3, 4, 6, 10.

Example 2 : State whether the following statements are true or false, and justify your answers. Solution : i This statement is true, because equilateral triangles have equal sides, and therefore are isosceles. Give a counter-example for it. This is false, because all integers are rational numbers. Example 4 : Restate the following statements with appropriate conditions, so that they become true statements: i If the diagonals of a quadrilateral are equal, then it is a rectangle.

Remark : There can ch 6 maths class 10 theorems kit other ways of restating the statements. For instance, iii can also be restated as p is irrational for all positive integers p which are not a perfect square. State whether the following statements are always true, always false or ambiguous.

State whether the following statements are true or false. Let a and b be real numbers such that ab 0. Then which of the following statements are true? Restate the following statements with appropriate conditions, so that they become true. Here, we will work with many more examples which will illustrate how deductive reasoning is.

The given statements are called premises or hypotheses. We ch 6 maths class 10 theorems kit with some examples. In which state does Shabana live? Solution : Here we have two premises: i Ch 6 maths class 10 theorems kit is in the state of Karnataka ii Shabana lives in Bijapur From these premises, we deduce that Shabana lives in the state of Karnataka.

Example 6 : Given that all mathematics textbooks are interesting, and suppose you are reading a mathematics textbook. What can we conclude about the textbook you are reading? Solution : Using the two premises or hypotheseswe can deduce that you are reading an interesting textbook.

What is y? What can you conclude about the lengths of DC and BC? So, we deduce that all the properties that hold for a parallelogram hold for ABCD.

Therefore, in particular, the property that the opposite sides of a parallelogram are equal to each other, holds. Remark : In this example, we have seen how we will often need to find out and use properties hidden in a given premise.

Example 9 : Given that p is irrational for all primes p, and suppose that is a prime. What can you conclude about ? Solution : We can conclude that is irrational. In the examples above, you might have noticed that we do not know whether the hypotheses are true or not.

We are assuming that they are true, and then applying deductive reasoning. For instance, in Example 9, we havent checked whether Fig.

What we are trying to emphasise in this section is that given a particular statement, how we use deductive reasoning to arrive at a conclusion. What really matters here is that we use the correct process of reasoning, and this process of reasoning does not depend on the trueness or falsity of the hypotheses. However, it must also be noted that if we start with an incorrect premise or hypothesiswe may arrive at a wrong conclusion.

Given that all women are mortal, and suppose that A is a woman, what can we conclude about A? Given that the product of two rational numbers ch 6 maths class 10 theorems kit rational, and suppose a and b are rationals, what can you conclude about ab?

Given that the decimal expansion of irrational numbers is non-terminating, non-recurring, and 17 is irrational, what can we conclude about the decimal expansion of 17? What can you conclude about the other angles of the parallelogram? Given that PQRS is a cyclic quadrilateral and also its diagonals bisect each. What can you conclude about the quadrilateral? Given that p is irrational for all primes p and also suppose that is a prime.

Can you conclude that is an irrational number? Is your conclusion correct? Why or why not? The first circle has one point on it, the second two ch 6 maths class 10 theorems kit, the third three, and so on.

All possible lines connecting the points ch 6 maths class 10 theorems kit drawn in each case. The lines divide the circle into mutually exclusive regions having no common portion. We can count these and tabulate our results as shown : Fig. From Class IX, you may remember that this intelligent guess is called a conjecture. Suppose your conjecture is that given n points on a circle, there are 2 n 1 mutually exclusive regions, created by joining the points with all possible lines.

Ch 6 maths class 10 theorems kit, having verified this formula for 5 points, are you satisfied that for any n points there are 2 n 1 regions? To deal with such questions, you would need a proof which shows beyond doubt that this result is true, or a counter- example to show that this result fails for some n.

This demonstrates the power of a counter-example. You may recall that in the Class IX we discussed that to disprove a statement, it is enough to come up with a single counter- example. Let us consider a few more examples. What we need is a proof which establishes its truth beyond doubt.

You shall learn a proof for the same in higher classes. You were not satisfied by only drawing several such figures, measuring the lengths of the respective tangents, and verifying for yourselves that the result was true in each case. Do you remember what did the proof consist of?

It consisted of a sequence of statements called valid argumentseach following from the earlier statements in the proof, or from previously proved and known results independent from the result to be proved, or from axioms, or from definitions, or from the assumptions you had. This is the way any proof is constructed. We shall now look at some examples and theorems and analyse their proofs to help us in getting a better understanding of how they are constructed.

We begin by using the so-called direct or deductive method of proof. In this method, we make several statements. Each is based on ch 6 maths class 10 theorems kit statements. If each statement is logically correct i.

Ch 6 maths class 10 theorems kit 10 : The sum of two ch 6 maths class 10 theorems kit numbers is a rational number. Solution : S. Let x and y be rational numbers. Since the result is about rationals, we start with x and y which are rational. Since n 0 and q 0, it follows that Using known properties of nq 0.

Remark : Note that, each statement in the proof above is based on a previously established fact, or definition. Let p be a prime number greater than 3. Since the result has to do with a prime number greater than ch 6 maths class 10 theorems kit, we start with such a number.

So, they are p is prime.

Abstract:

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All congruent figures are similar but the similar figures need not be congruent. Area Theorem: The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Pythagoras theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Visit to Discussion Forum to share your knowledge. This platform is being maintained to discuss the doubts of all students with experts and teachers. For any inconvenience, please contact us for help.

We will try to solve your difficulty as soon as possible. What is meant by Similarity or Similar Triangle? What is Area Theorem in Class 10? State Pythagoras theorem? State Converse of Pythagoras theorem. To verify and use results given in the curriculum based on similarity theorems.

There are so many other important theorems based on similar triangles like If in two triangles, sides of one triangle are proportional to i. Or if two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.

Historical Facts! Pythagoras theorem is famous because of its wide range of applications. Thales of Miletus � BC, Greece was the first known philosopher and mathematician. He is credited with the first use of deductive reasoning in geometry. He discovered many propositions in geometry. He is believed to have found the heights of the pyramids in Egypt, using shadows and principle of similar triangles.

Height of pyramids can also be find using applications of trigonometry. That is, from the smallest value to the highest value. Median is calculated as. Where n is the number of values in the data. If the number of values in the data set is even, then the median is the average of the two middle values. Mode: Mode of a statistical data is the value of that variable which has the maximum frequency. Median: For the given data, we need to have class interval, frequency distribution and cumulative frequency distribution.

Then, median is calculated as. Mode: Modal class : The class interval having highest frequency is called the modal class and Mode is obtained using the modal class. Understanding the basic concepts and learning all the important formulas is extremely sufficient to pass the Maths exam with flying colours.

If you know the formulas very well then it will not take much time for you to solve questions in the exam paper. So, keep practicing with the list of important formulas given above in this article. Jagranjosh Education Awards Click here if you missed it! This website uses cookie or similar technologies, to enhance your browsing experience and provide personalised recommendations.

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