RS Aggarwal Solutions: Introduction to Euclid's Geometry | Mathematics (Maths) Class 9 PDF Download

 Page 1


Question:1
What is the difference between a theorem and an axiom?
Solution:
An axiom is a basic fact that is taken for granted without proof.
Examples:
i) Halves of equals are equal.
ii) The whole is greater than each of its parts.
Theorem: A statement that requires proof is called theorem. 
Examples:
i) The sum of all the angles around a point is 360
°
.
ii) The sum of all the angles of triangle is 180
°
.        
Question:2
Define the following terms:
i
Line segment
ii
Ray
iii
Intersecting lines
iv
Parallel lines
v
Half line
vi
Concurrent lines
vii
Collinear points
viii
Plane
Solution:
i Line segment :A line segment is a part of line that is bounded by two distinct end-points. A line segment has a
fixed length.
i i Ray:  A line with a start point but no end point and without a definite length is a ray.
i i i Intersecting lines: Two lines with a common point are called intersecting lines.
 
Page 2


Question:1
What is the difference between a theorem and an axiom?
Solution:
An axiom is a basic fact that is taken for granted without proof.
Examples:
i) Halves of equals are equal.
ii) The whole is greater than each of its parts.
Theorem: A statement that requires proof is called theorem. 
Examples:
i) The sum of all the angles around a point is 360
°
.
ii) The sum of all the angles of triangle is 180
°
.        
Question:2
Define the following terms:
i
Line segment
ii
Ray
iii
Intersecting lines
iv
Parallel lines
v
Half line
vi
Concurrent lines
vii
Collinear points
viii
Plane
Solution:
i Line segment :A line segment is a part of line that is bounded by two distinct end-points. A line segment has a
fixed length.
i i Ray:  A line with a start point but no end point and without a definite length is a ray.
i i i Intersecting lines: Two lines with a common point are called intersecting lines.
 
i v Parallel lines: Two lines in a plane without a common point are parallel lines.
 
v Half line: A straight line extending from a point indefinitely in one direction only is a half line.
 
v i Concurrent lines: Three or more lines intersecting at the same point are said to be concurrent.
 
v i i Collinear points: Three or more than three points are said to be collinear if there is a line, which contains all the
points.
 
v i i i Plane: A plane is a surface such that every point of the line joining any two point on it, lies on it.
Question:3
In the adjoining figure, name
i
six points
ii
five lines segments
iii
four rays
iv
four lines
v
four collinear points
Solution:
Page 3


Question:1
What is the difference between a theorem and an axiom?
Solution:
An axiom is a basic fact that is taken for granted without proof.
Examples:
i) Halves of equals are equal.
ii) The whole is greater than each of its parts.
Theorem: A statement that requires proof is called theorem. 
Examples:
i) The sum of all the angles around a point is 360
°
.
ii) The sum of all the angles of triangle is 180
°
.        
Question:2
Define the following terms:
i
Line segment
ii
Ray
iii
Intersecting lines
iv
Parallel lines
v
Half line
vi
Concurrent lines
vii
Collinear points
viii
Plane
Solution:
i Line segment :A line segment is a part of line that is bounded by two distinct end-points. A line segment has a
fixed length.
i i Ray:  A line with a start point but no end point and without a definite length is a ray.
i i i Intersecting lines: Two lines with a common point are called intersecting lines.
 
i v Parallel lines: Two lines in a plane without a common point are parallel lines.
 
v Half line: A straight line extending from a point indefinitely in one direction only is a half line.
 
v i Concurrent lines: Three or more lines intersecting at the same point are said to be concurrent.
 
v i i Collinear points: Three or more than three points are said to be collinear if there is a line, which contains all the
points.
 
v i i i Plane: A plane is a surface such that every point of the line joining any two point on it, lies on it.
Question:3
In the adjoining figure, name
i
six points
ii
five lines segments
iii
four rays
iv
four lines
v
four collinear points
Solution:
i
Points are A, B, C, D, P and R.
ii
 
¯
EF, 
¯
GH, 
¯
FH , 
¯
EG, 
¯
MN
iii
 
?
EP, 
?
GR, 
?
HS, 
?
FQ
iv
 
?
AB, 
?
CD, 
?
PQ, 
?
RS
v
Collinear points are M, E, G and B.
Question:4
In the adjoining figure, name:
i
two pairs of intersecting lines and their corresponding points of intersection
ii
three concurrent lines and their points of intersection
iii
three rays
iv
two line segments
Solution:
i
Two pairs of intersecting lines and their point of intersection are
?
EF, 
?
GH, point R , 
?
AB, 
?
CD, point P
ii
Three concurrent lines are
?
AB, 
?
EF, 
?
GH, point R
iii
Three rays are
?
RB, 
?
RH, 
?
RF
{ } { }
{ }
{ }
Page 4


Question:1
What is the difference between a theorem and an axiom?
Solution:
An axiom is a basic fact that is taken for granted without proof.
Examples:
i) Halves of equals are equal.
ii) The whole is greater than each of its parts.
Theorem: A statement that requires proof is called theorem. 
Examples:
i) The sum of all the angles around a point is 360
°
.
ii) The sum of all the angles of triangle is 180
°
.        
Question:2
Define the following terms:
i
Line segment
ii
Ray
iii
Intersecting lines
iv
Parallel lines
v
Half line
vi
Concurrent lines
vii
Collinear points
viii
Plane
Solution:
i Line segment :A line segment is a part of line that is bounded by two distinct end-points. A line segment has a
fixed length.
i i Ray:  A line with a start point but no end point and without a definite length is a ray.
i i i Intersecting lines: Two lines with a common point are called intersecting lines.
 
i v Parallel lines: Two lines in a plane without a common point are parallel lines.
 
v Half line: A straight line extending from a point indefinitely in one direction only is a half line.
 
v i Concurrent lines: Three or more lines intersecting at the same point are said to be concurrent.
 
v i i Collinear points: Three or more than three points are said to be collinear if there is a line, which contains all the
points.
 
v i i i Plane: A plane is a surface such that every point of the line joining any two point on it, lies on it.
Question:3
In the adjoining figure, name
i
six points
ii
five lines segments
iii
four rays
iv
four lines
v
four collinear points
Solution:
i
Points are A, B, C, D, P and R.
ii
 
¯
EF, 
¯
GH, 
¯
FH , 
¯
EG, 
¯
MN
iii
 
?
EP, 
?
GR, 
?
HS, 
?
FQ
iv
 
?
AB, 
?
CD, 
?
PQ, 
?
RS
v
Collinear points are M, E, G and B.
Question:4
In the adjoining figure, name:
i
two pairs of intersecting lines and their corresponding points of intersection
ii
three concurrent lines and their points of intersection
iii
three rays
iv
two line segments
Solution:
i
Two pairs of intersecting lines and their point of intersection are
?
EF, 
?
GH, point R , 
?
AB, 
?
CD, point P
ii
Three concurrent lines are
?
AB, 
?
EF, 
?
GH, point R
iii
Three rays are
?
RB, 
?
RH, 
?
RF
{ } { }
{ }
{ }
iv
Two line segments are
¯
RQ and 
¯
RP
Question:5
From the given figure, name the following:
a
Three lines
b
One rectilinear figure
c
Four concurrent points
Solution:
a
 Line 
?
PQ
, Line
?
RS
and Line
?
AB
b
 CEFG
c
No point is concurrent.
Question:6
i
How many lines can be drawn through a given point?
ii
How many lines can be drawn through two given points?
iii
At how many points can two lines at the most intersect?
iv
If A, B and C are three collinear points, name all the line segments determined by them.
Solution:
i
Infinite lines can be drawn through a given point.
{ }
Page 5


Question:1
What is the difference between a theorem and an axiom?
Solution:
An axiom is a basic fact that is taken for granted without proof.
Examples:
i) Halves of equals are equal.
ii) The whole is greater than each of its parts.
Theorem: A statement that requires proof is called theorem. 
Examples:
i) The sum of all the angles around a point is 360
°
.
ii) The sum of all the angles of triangle is 180
°
.        
Question:2
Define the following terms:
i
Line segment
ii
Ray
iii
Intersecting lines
iv
Parallel lines
v
Half line
vi
Concurrent lines
vii
Collinear points
viii
Plane
Solution:
i Line segment :A line segment is a part of line that is bounded by two distinct end-points. A line segment has a
fixed length.
i i Ray:  A line with a start point but no end point and without a definite length is a ray.
i i i Intersecting lines: Two lines with a common point are called intersecting lines.
 
i v Parallel lines: Two lines in a plane without a common point are parallel lines.
 
v Half line: A straight line extending from a point indefinitely in one direction only is a half line.
 
v i Concurrent lines: Three or more lines intersecting at the same point are said to be concurrent.
 
v i i Collinear points: Three or more than three points are said to be collinear if there is a line, which contains all the
points.
 
v i i i Plane: A plane is a surface such that every point of the line joining any two point on it, lies on it.
Question:3
In the adjoining figure, name
i
six points
ii
five lines segments
iii
four rays
iv
four lines
v
four collinear points
Solution:
i
Points are A, B, C, D, P and R.
ii
 
¯
EF, 
¯
GH, 
¯
FH , 
¯
EG, 
¯
MN
iii
 
?
EP, 
?
GR, 
?
HS, 
?
FQ
iv
 
?
AB, 
?
CD, 
?
PQ, 
?
RS
v
Collinear points are M, E, G and B.
Question:4
In the adjoining figure, name:
i
two pairs of intersecting lines and their corresponding points of intersection
ii
three concurrent lines and their points of intersection
iii
three rays
iv
two line segments
Solution:
i
Two pairs of intersecting lines and their point of intersection are
?
EF, 
?
GH, point R , 
?
AB, 
?
CD, point P
ii
Three concurrent lines are
?
AB, 
?
EF, 
?
GH, point R
iii
Three rays are
?
RB, 
?
RH, 
?
RF
{ } { }
{ }
{ }
iv
Two line segments are
¯
RQ and 
¯
RP
Question:5
From the given figure, name the following:
a
Three lines
b
One rectilinear figure
c
Four concurrent points
Solution:
a
 Line 
?
PQ
, Line
?
RS
and Line
?
AB
b
 CEFG
c
No point is concurrent.
Question:6
i
How many lines can be drawn through a given point?
ii
How many lines can be drawn through two given points?
iii
At how many points can two lines at the most intersect?
iv
If A, B and C are three collinear points, name all the line segments determined by them.
Solution:
i
Infinite lines can be drawn through a given point.
{ }
ii
Only one line can be drawn through two given points.
iii
 At most two lines can intersect at one point.
iv
The line segments determined by three collinear points A, B and C are
AB, 
¯
BC and AC.
Question:7
Which of the following statements are true?
i
A line segment has no definite length.
ii
A ray has no end-point.
iii
A line has a definite length.
iv
A line 
?
AB
is same as line 
?
BA
.
v
A ray ? AB
is same as ray ? BA
.
vi
Two distinct points always determine a unique line.
vii
Three lines are concurrent if they have a common point.
viii
Two distinct lines cannot have more than one point in common.
ix
Two intersecting lines cannot be both parallel to the same line.
x
Open half-line is the same thing as ray.
xi
Two lines may intersect in two points.
xii
Two lines are parallel only when they have no point in common.
Solution:
i
False. A line segment has a definite length.
ii
False. A ray has one end-point.