CHAPTER 6
LINES AND ANGLES
6.1 Introduction
In Chapter 5, you have studied that a minimum of two points are required to draw a
line. You have also studied some axioms and, with the help of these axioms, you
proved some other statements. In this chapter, you will study the properties of the
angles formed when two lines intersect each other, and also the properties of the
angles formed when a line intersects two or more parallel lines at distinct points.
Further you will use these properties to prove some statements using deductive reasoning
(see Appendix 1). You have already verified these statements through some activities
in the earlier classes.
In your daily life, you see different types of angles formed between the edges of
plane surfaces. For making a similar kind of model using the plane surfaces, you need
to have a thorough knowledge of angles. For instance, suppose you want to make a
model of a hut to keep in the school exhibition using bamboo sticks. Imagine how you
would make it? You would keep some of the sticks parallel to each other, and some
sticks would be kept slanted. Whenever an architect has to draw a plan for a multistoried
building, she has to draw intersecting lines and parallel lines at different angles. Without
the knowledge of the properties of these lines and angles, do you think she can draw
the layout of the building?
In science, you study the properties of light by drawing the ray diagrams.
For example, to study the refraction property of light when it enters from one medium
to the other medium, you use the properties of intersecting lines and parallel lines.
When two or more forces act on a body, you draw the diagram in which forces are
represented by directed line segments to study the net effect of the forces on the
body. At that time, you need to know the relation between the angles when the rays
(or line segments) are parallel to or intersect each other. To find the height of a tower
or to find the distance of a ship from the light house, one needs to know the angle
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formed between the horizontal and the line of sight. Plenty of other examples can be
given where lines and angles are used. In the subsequent chapters of geometry, you
will be using these properties of lines and angles to deduce more and more useful
properties.
Let us first revise the terms and definitions related to lines and angles learnt in
earlier classes.
6.2 Basic Terms and Definitions
Recall that a part (or portion) of a line with two end points is called a line-segment
and a part of a line with one end point is called a ray. Note that the line segment AB is
denoted by AB , and its length is denoted by AB. The ray AB is denoted by AB , and
a line is denoted by AB . However, we will not use these symbols, and will denote
the line segment AB, ray AB, length AB and line AB by the same symbol, AB. The
meaning will be clear from the context. Sometimes small letters l, m, n, etc. will be
used to denote lines.
If three or more points lie on the same line, they are called collinear points;
otherwise they are called non-collinear points.
Recall that an angle is formed when two rays originate from the same end point.
The rays making an angle are called the arms of the angle and the end point is called
the vertex of the angle. You have studied different types of angles, such as acute
angle, right angle, obtuse angle, straight angle and reflex angle in earlier classes
(see Fig. 6.1).
(i) acute angle : 0° < x < 90° (ii) right angle : y = 90° (iii) obtuse angle : 90° < z < 180°
(iv) straight angle : s = 180° (v) reflex angle : 180° < t < 360°
Fig. 6.1 : Types of Angles
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LINES AND ANGLES
6.1 Introduction
In Chapter 5, you have studied that a minimum of two points are required to draw a
line. You have also studied some axioms and, with the help of these axioms, you
proved some other statements. In this chapter, you will study the properties of the
angles formed when two lines intersect each other, and also the properties of the
angles formed when a line intersects two or more parallel lines at distinct points.
Further you will use these properties to prove some statements using deductive reasoning
(see Appendix 1). You have already verified these statements through some activities
in the earlier classes.
In your daily life, you see different types of angles formed between the edges of
plane surfaces. For making a similar kind of model using the plane surfaces, you need
to have a thorough knowledge of angles. For instance, suppose you want to make a
model of a hut to keep in the school exhibition using bamboo sticks. Imagine how you
would make it? You would keep some of the sticks parallel to each other, and some
sticks would be kept slanted. Whenever an architect has to draw a plan for a multistoried
building, she has to draw intersecting lines and parallel lines at different angles. Without
the knowledge of the properties of these lines and angles, do you think she can draw
the layout of the building?
In science, you study the properties of light by drawing the ray diagrams.
For example, to study the refraction property of light when it enters from one medium
to the other medium, you use the properties of intersecting lines and parallel lines.
When two or more forces act on a body, you draw the diagram in which forces are
represented by directed line segments to study the net effect of the forces on the
body. At that time, you need to know the relation between the angles when the rays
(or line segments) are parallel to or intersect each other. To find the height of a tower
or to find the distance of a ship from the light house, one needs to know the angle
2024-25
, 70 MATHEMATICS
formed between the horizontal and the line of sight. Plenty of other examples can be
given where lines and angles are used. In the subsequent chapters of geometry, you
will be using these properties of lines and angles to deduce more and more useful
properties.
Let us first revise the terms and definitions related to lines and angles learnt in
earlier classes.
6.2 Basic Terms and Definitions
Recall that a part (or portion) of a line with two end points is called a line-segment
and a part of a line with one end point is called a ray. Note that the line segment AB is
denoted by AB , and its length is denoted by AB. The ray AB is denoted by AB , and
a line is denoted by AB . However, we will not use these symbols, and will denote
the line segment AB, ray AB, length AB and line AB by the same symbol, AB. The
meaning will be clear from the context. Sometimes small letters l, m, n, etc. will be
used to denote lines.
If three or more points lie on the same line, they are called collinear points;
otherwise they are called non-collinear points.
Recall that an angle is formed when two rays originate from the same end point.
The rays making an angle are called the arms of the angle and the end point is called
the vertex of the angle. You have studied different types of angles, such as acute
angle, right angle, obtuse angle, straight angle and reflex angle in earlier classes
(see Fig. 6.1).
(i) acute angle : 0° < x < 90° (ii) right angle : y = 90° (iii) obtuse angle : 90° < z < 180°
(iv) straight angle : s = 180° (v) reflex angle : 180° < t < 360°
Fig. 6.1 : Types of Angles
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