NCERT Mathematics Class 9 Exemplar Ch 5 Introduction to Euclid's Geometry Part 8

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Exercise 5.4

Q.1. An equilateral triangle is a polygon made up of three line segments out of which two line segments are equal to the third one and all its angles are 60° each. Define the terms used in this definition which you feel necessary. Are there any undefined terms in this? Can you justify that all sides and all angles are equal in an equilateral triangle?

Answer: The terms need to be defined are

  • Polygon is a closed figure bounded by three or more line segments.

  • Line segment is part of a line with two end points.

  • Line undefined term.

  • Point Undefined term.

  • Angle in a figure is formed by two rays with one common initial point.

  • Acute angle is an angle whose measure is between

Here undefined terms are line and point. All the angles of equilateral triangle are 600 each (given) Two line segments are equal to the third-one (given) Therefore, all three sides of an equilateral triangle are equal (according to Euclid’s axiom, things which are equal to the same thing are equal to one another).

Q.2. Study the following statement: “Two intersecting lines cannot be perpendicular to the same line”. Check whether it is an equivalent version to the Euclid’s fifth postulate. [Hint: Identify the two intersecting lines l and m and the line n in the above statement.]

Answer: Two equivalent versions of Euclid’s fifth postulate are

  • For every line L and for every point P not lying on L, there exists a unique line M passing through P and parallel to L.

  • Two distinct intersecting lines cannot be parallel to the same line.

From above two segments it is clear that given statement is not an equivalent version to the Euclid’s fifth postulate.

Q.3. Read the following statements which are taken as axioms:

(i) If a transversal intersects two parallel lines, then corresponding angles are not necessarily equal.

(ii) If a transversal intersects two parallel lines, then alternate interior angles are equal. Is this system of axioms consistent? Justify your answer.

Answer:

  • A system of axiom is called consistent, if there is no statement which can be deduced from these axioms such that it contradicts any axiom. We know that, if a transversal intersects two parallel lines, then each pair of corresponding angles are equal, which is a theorem.

  • So, Statement (i) is false and not an axiom. Also, we know that, if a transversal intersects two parallel lines, then each pair of alternate interior angles are equal. It is also a theorem. So, Statement (ii) is true and an axiom.

  • Thus, in given statements, first is false and second is an axiom. Hence. Given system of axioms is not consistent.

Q.4. Read the following two statements which are taken as axioms:

(i) If two lines intersect each other, then the vertically opposite angles are not equal.

(ii) If a ray stands on a line, then the sum of two adjacent angles so formed is equal to 180°. Is this system of axioms consistent? Justify your answer.

Answer:

  • A system of axiom is called consistent, if there is no statement which can be deduced from these axioms such that it contradicts any axiom. We know that, if two lines intersect each other, then the vertically opposite angles are equal. It is a theorem, so given statement (i) is false and not an axiom.

  • Also, we know that, if a ray stands on line, then the sum of two adjacent angles so formed is equal to 180°. It is an axiom. So, given statement (ii) is true and an axiom. Thus, in given statements, first is false and second is true. Hence, given system of axioms is not consistent.

Q.5. Read the following axioms:

(i) Things which are equal to the same thing are equal to one another.

(ii) If equals are added to equals, the wholes are equal.

(iii) Things which are double of the same thing are equal to one another. Check whether the given system of axioms is consistent or inconsistent.

Answer: Some of the Euclid’s axioms are

  • The things which are equal to the same thing are equal to one another.

  • If equals be added to the equals, the wholes are equal.

  • If equals be subtracted from equals, the remainders are equals.

  • Things which coincide with one another are equal to one another.

  • The whole is greater than the part.

  • Things which are double of the same thing are equal to one another.

  • Things which are halves of the same thing are equal to one another.

Given axioms are: (i) Things which are equal to the same thing are equal to one another. (ii) If equals are added to equals, the wholes are equal. (iii) Things which are double of the same things are equal to one another. Thus, given three axioms are Euclid’s axioms which do not contradict any axioms. So, given system of axioms is consistent.

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