NCERT Solutions for Class 9 Math Chapter 5 - Introduction to Euclids Geometry

Answer:

False
Correct statement:
Infinitely many lines can pass through a single point.
This is self evident and can be seen visually by the student as follows:

Answer:

False, because the given statement contradicts the postulate I of the Euclid that assures that there is a unique line that passes through two distinct points.

Through two points P and Q a unique line can be drawn.

Answer:

True
Evidence: According to Euclid’s postulate 2; A terminated line can be produced indefinitely.

Answer:

True
Evidence:
According to the axiom 4 of Euclid; that “the things which coincide with one another are equal to one another”. If we superimpose the region bounded by one circle on the other, then they coincide. So, their centres and boundaries coincide. Therefore, their radii will coincide.

Answer:

True.
Evidence: Euclid’s axiom 1 states that the things which are equal to the same thing are equal to one another.

Answer:

Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.
Other term involved is the “plane”. We keep the plane as undefined term. The only thing is that we can represent it intuitively or explain it with the help of physical model.

Answer:

Perpendicular lines. When a straight line set up on a straight line makes the adjacent angles equal to one another, each of equal angle is right and the straight line standing on the other is called a perpendicular to that on which it stands. The other terms that need to be defined first is (1) angle (2) adjacent angles (3) right angle. Let us define these:
(1) Angle. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.
(2) Adjacent angles. The two angles with the same vertex, one arm common and other arms lying on the opposite sides of the common arm are called adjacent angles.
(3) Right angle. An angle equal to one quarter of a complete angle is called a right angle.

Answer:

Line segment. A line segment which extends indefinitely in both irections gives a line.
Other term involved is line. We keep the line as undefined term. The only thing is that we can represent it intuitively or explain it with the help of physical model.

Answer:

Radius of a circle. A line segment joining the centre to any point on the circle is called the radius of the circle.
Other terms that need to be defined first are:
(1) Circle (2) Centre.
Let us define these:
(1) Circle. A circle is a closed curve on a plane, all points on which are at the same distance from a fixed point within it.
(2) Centre of circle. The fixed point from which all the points on a circle are equidistant is called its centre.

Answer:

Square. Of quadrilateral figures, a square is that which is both equilateral and rightangled. Other terms that need to be defined first are (1) equilateral (2) right angle. Let us define these.
(1) Equilateral. A figure having all its sides equal is called an equilateral.
(2) Right angle. An angle equal to one quarter of a complete angle is called a right angle.

Answer:

There are several undefined terms which are to be listed by a student. These two postulates (i) and (ii) are consistent because they deal with two different situations.
Postulate (i) states that given two points A and B, there is a point C lying on the line in between them.

Postulate (ii) states that for given two points A and B, we can take point C not lying on the line through A and B.

Hence we observe that the postulates do not follow from Euclid’s postulate. However they follow from Euclid’s axiom that follow from Euclid’s postulate (1) that “for given two distinct points there is a unique line that passes through them.” See fig.

Answer:

Answer:

Let C and D be the two mid-points of line
segment AB.
So, according to Euclid’s axiom (4) when line is folded about point C we observe that part BC superimposes over the part AC.
It implies that
AC = BC ... (i)
Similarly, D is the mid-point of AB implies that
AD = BD ... (ii)
we have AB = AB
or AC + BC = AD + BD
or AC + AC = AD + AD
[Using (i) and (ii)]
or 2AC = 2AD
or AC = AD ... (iii)

When we superimpose AD over AC and BD over BC we find that D exactly lies over C. It implies that D and C are not two different points but the same.
Hence, we conclude that mid-point of a line segment is unique.

Answer:

Given that
AC = BD ... (i)
AC = AB + BC
[Point B lies between A and C.] ... (ii)
BD = BC + CD
[Point C lies between B and D.] ...(iii)
Substituting (ii) and (iii) in (i), we get:
AB + BC = BC + CD
Subtracting BC from both sides, we get:
AB + BC – BC = BC + CD – BC
= BC – BC + CD
So, AB = CD
[_ if equals are subtracted from equals, the remaining are equal.]

Answer:

Euclid’s Axiom 5 states that “The whole is greater than the part.”
Since this is true for anything in any part of the world.
So, this is a universal truth.

Answer:

There are several easy equivalent versions of Euclid’s fifth postulate. Playfair’s axiom is one of the other equivalent versions of Euclid’s fifth postulate which is easily understandable. According to it,

For every line l and for every point not lying on l, there exists a unique line m passing through point P and is parallel to l.
Clearly, if all lines passing through the point P, only line m is parallel to line l.

Answer:

Yes. Euclid’s fifth postulate is valid for parallelism of lines because if a straight line l falls on two straight lines m and n such that sum of the interior angles on one side of l is two right angles, then by Euclid’s fifth postulate the line will not meet on this side of l. Next, you know that the sum of the interior angles on the other side of line l will also be two right angles. Therefore, they will not meet on the other side also. So, the lines m and n never meet and are, therefore, parallel.