Basic Definitions - Lines and Angles, Class 9, Mathematics PDF Download

INTRODUCTION
In this chapter, we 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.

BASIC TERMS AND DEFINITIONS

(a) LINE-SEGMENT :– A part or portion of a line with two end points is called a line-segment. The line segment AB is denoted by and its length is denoted by AB.

(b) RAY :– A part of a line with one end point is called a ray. Ray AB is denoted by .

(c) LINE :– A line is the collection of infinite number of points and extends endlessly in both the directions. A line is generally denoted by small letters such as ,m,n,p,q,r etc.

(d) COLLINEAR POINTS :– If three or more points lie on the same line, then they are called collinear points. Points A, B, C, D, E, P and Q are collinear points because they all lie on the same line .

(e) NON-COLLINEAR POINTS :– If three or more points does not lie on the same line, they are called non-collinear points.
In figure, we observe that points A, B, C, E and P are collinear points but the points. A, B, C, D, E and P are not collinear. Similarly A, B, C, D, E, P and Q are also not collinear.

(f) INTERSECTING LINES :– Two distinct lines are intersecting, if they have a common point. The common point is called the "point of intersection" of the two lines. We observe in figure, the intersection point of two intersecting lines is a unique point.

NON-INTERSECTING LINES (PARALLEL LINES) :– Two distinct lines which are not intersecting are said to be parallel lines. The parallel lines are always at a constant distance from each other. If figure, the lines PQ and RS do not intersect and hence the lines PQ and RS are parallel to each other.

ANGLE :– 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.

Remark : Every angle has a measure and unit of measurement is degree.
1 right angle = 90°
1° = 60'(minutes), 1 minute = 60" (second)
The angles are of following types :–

(i) Acute angle :– An angle whose measure is less than 90° is called an acute angle.

(ii) RIGHT ANGLE :– An angle whose measure is 90° is called a right angle.

(iii) OBTUSE ANGLE :– An angle whose measure is more than 90° but less than 180° is called obtuse angle.

(iv) STRAIGHT ANGLE :– An angle whose measure is 180° is called a straight angle.

(v) REFLEX ANGLE :- An angle whose measure is more than 180° but less than 360° is called a reflex

(vi) COMPLEMENTARY ANGLES :- Two angles, the sum of whose measures is 90° are called complementary angles.

∠AOC & ∠BOC are complementary angles, as x° + y° = 90°

For example : Let, ∠AOC = 35° and ∠BOC = 55°

∴ ∠AOC + ∠BOC = 35° + 55° = 90°

Therefore ∠AOC and ∠BOC are complementary to each other.

(vii) SUPPLEMENTARY ANGLES :- Two angles, the sum of whose measures is 180° are called supplementary angles.

∠AOC & ∠BOC are supplementary angle, as x° + y° = 180°.
For example : Let, ∠AOC = 47° and ∠BOC = 133°
∴ ∠AOC + ∠BOC = 47° + 133° = 180°
Therefore ∠AOC and ∠BOC are supplementary to each other.

(viii) ADJACENT ANGLES :- Two angles are called adjacent angles, if :

(i) they have the same vertex.
(ii) they have a common arm, and
(iii) uncommon arms are on either side of the common arm.
In figure, ∠AOX and ∠BOX are adjacent angles. OX is common arm, OA and OB are non-common arms lies on the either side of OX.

When two angles are adjacent, then their sum is always equal to the angle formed by the two non-common arms. So, we can write,
∠AOB = ∠AOX + ∠BOX

(ix) LINEAR PAIR OF ANGLES :- Two adjacent angles are said to form a linear pair of angles, if non-common arms are two opposite rays.

In figure, the ∠BOC and ∠COA form a lineOar pair of angles because the non-common arms OA and OB are two opposite rays.
∠AOC + ∠BOC = ∠AOB [∵ ∠AOC and ∠BOC are adjacent angles]
∴ ∠AOC + ∠BOC = 180° [Straight angle = 180°]