Exercise 5.1
(1) Which of the following statements are true and which are false? Give reasons for your answers.
(i) Only one line can pass through a single point.
(ii) There are an infinite number of lines which pass through two distinct points.
(iii) A terminated line can be produced indefinitely on both the sides.
(iv) If two circles are equal, then their radii are equal.
(v) In fig below, if AB = PQ and PQ = XY, then AB = XY.
Ans-
(i) Only one line can pass through a single point.
It is a false statement because many lines can be pass through a point. Only one line can pass through point A to point B.
(ii) There are an infinite number of lines which pass through two distinct points.
It is a false statement because only one line can pass through point A to point B.
(iii) A terminated line can be produced indefinitely on both the sides.
It is a true statement because terminated line is a line segment which can be produced indefinitely on both the sides.
(iv) If two circles are equal, then their radii are equal.
It is a true statement because two equal circles can be drawn by equal radii.
(v) In fig below, if AB = PQ and PQ = XY, then AB = XY.
It is a true statement because things which are equal to the same thing are equal to one another.
(2) Give a definition for each of the following terms. Are there other terms that need to be defined first? What are they, and how might you define them?
(i) parallel lines (ii) perpendicular lines (iii) line segment (iv) radius of a circle (v) square
Ans-
(i) Definition- Parallel lines are lines at the equal distance.
Lines should be defined first.
Terms should be define first are given below –
A line segment when extends its both end points in either direction endlessly, is called a line.
(ii) Definition-When one line makes an angle of 900 to another line, these lines are called as perpendicular lines.
Angle, lines should be defined first.
Terms should be define first are given below –
(a) The distance between two rays or two line segments or two lines diverging from vertex (common point) is called an angle.
(b) A line segment when extends its both end points in either direction endlessly, is called a line.
(iii) Definition –Line segment is a segment which can be produced indefinitely on both the sides to make it a line.
Line segment and line should be defined first.
Terms should be define first are given below –
(a) A part of a line that has two end points is called line segment.
(b) A line segment when extends its both end points in either direction endlessly, is called a line.
(iv) Definition – The radius of a circle is a line segment from its center to its perimeter.
Circle, line segment, center of the circle, radius and perimeter of the circle should be defined.
Terms should be define first are given below –
(a) A part of a line that has two end points is called line segment.
(b) A circle is a round plane figure whose boundary has infinite points equidistant from a fixed point.
(c) The center of a circle is a given point from which all points of circles are at equidistance from this fix point.
(d) The radius of a circle is a line segment from its center to its perimeter.
(e) Perimeter of the circle is a boundary or distance around a circle.
(v) Definition – Square is a closed figure of equal four sides which meets at right angle. Opposite sides are parallel to each other.
Closed figure, sides, right angle, opposites sides and parallel sides should be defined.
Terms should be define first are given below –
(a) A closed figure is a figure which has same starting and end point.
(b) Sides are arms or line segment of the square.
(c) When two lines meet at 900, is called right angle.
(d) Opposite sides are those lines of a closed figure which do not have common same end point.
(e) Parallel lines are those lines which are at equidistance and never meet.
(3) Consider two “postulates” given below:
(i) Given any two distinct points A and B, there exists a third point C which is in between A and B.
(ii) There exist at least three points that are not on the same line.
Do these postulates contain any undefined terms? Are these postulates consistent?
Do they follow from Euclid’s postulates? Explain.
Ans-
Yes, points and line should be defined.
Yes, these postulates are consistent because these postulates do not make a statement that contradicts any axiom.
They do not follow Euclid postulates but follow axiom 5.1.
Axiom 5.1- Given two distinct points, there is a unique line that passes through them.
The above axiom is followed in both given postulates because
(i) Line AB is passes through Point A and B.
(ii) Three different points do not exist on the same line.
(4) If a point C lies between two points A and B such that AC = BC, then prove that AC = AB. Explain by drawing the figure.
Ans-
Point C lies in between point A and point B.
AC = BC
AB = AC + BC
= AC + AC
AB = 2 AC
AC = AB
Hence, proved.
(5) In question 4, point C is called a mid – point of line – segment AB. Prove that every line segment has one and only one mid – point.
Ans-
C is mid point of line AB.
AC = CB
AB = AC + CB
Let D is another mid point of the line AB.
AD = DB
AB = AD + DB
Things which are equal to the same thing are equal to one another.
Therefore,
AC + CB = AD + DB
If equals are added to equals, the wholes are equals. Therefore
AC + CB + AC = AD + DB + AC
2AC = AD + DB + AC – CB
= AD + DB + AC – AC
= AB
Similarly
AD + DB = AC + CB
If equals are added to equals, the wholes are equals. Therefore
AD + CB + AD = AC + CB + AD
2AD = AC + CB + AD – CB
= AB + CB _ Cb
2AD = AB
2AC = 2AD
Things which are double of the same things are equal to one another.
Therefore,
AC = AD
Things which coincide with one another are equal to one another.
Therefore, point C and point D are same points.
(6) In figure below, if AC = BD, then prove that AB = CD.
Ans-
AC = BD (Given)
AC = AB + BC
BD = BC + CD
Things which are equal to the same thing are equal to one another.
Therefore,
AB + BC = BC + CD
If equals are subtracted from equals, the remainders are equals.
Therefore,
AB + BC – BC = BC + CD – BC
AB = CD
Hence, proved.
(7) Why is Axiom 5, in the list of Euclid’s axioms, considered a ‘universal truth’? (Note that the question is not about the fifth postulate).
Ans- Axiom 5 of Euclid – The whole is greater than the part.
It is considered a ‘universal truth’ because it is always true in any condition and any where of world.
Eg:-
(i)A book has 100 pages, and then it is considered as a whole. But, if we take out 10 pages from the book, then these pages (parts) are lesser then the book as a whole.
(ii) A pizza which has 8 parts is considered as a whole. But if we take out or eat 2 parts of pizza. The two parts are lesser than the pizza as a whole.
Exercise 5.2
(1) How would you rewrite Euclid’s fifth postulate so that it would be easier to understand?
Ans- If a straight line cut the two straight lines the sum of the interior angles on the same side is measure less than 1800 and if these straight lines extended they will meet on the side of the interior angles whose sum are less than 1800 .
(2) Does Euclid’s fifth postulate imply the existence of parallel lines? Explain.
Ans-
Euclid’s fifth postulate does not imply the existence of parallel lines because parallel lines never meets but according to the postulates two straight lines can meet, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.
Euclid’s fifth postulate can imply the existence of the parallel line if sum of the interior angles should be equal to sum of two right angles. It should be true for other sides of interior angles of transversal line.
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