NCERT Class 9 Mathematics Textbook for Blind Students made Screen Readable by Professor T K Bansal.

Line, ray, and line segment

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 (dash on AB). The ray AB is denoted by AB with → on AB, and a line is denoted by AB with ↔ on 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. are used to denote lines.

Collinear and non-collinear points

If three or more points lie on the same line, they are called collinear points; otherwise, they are called non-collinear points.

Angles

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. In our earlier classes, we have studied different types of angles such as, acute angle, right angle, obtuse angle, straight angle and reflex angle(see Figure 6.1).

Figure 6.1(i) : Acute Angle

Figure 6.1(ii) : Right Angle

Figure 6.1(iii) : Obtuse Angle

Figure 6.1(iv) : Straight Angle

Figure 6.1(v) : Reflex Angle

An acute angle measures between 0° and 90°, whereas a right angle is exactly equal to 90°. An angle greater than 90° but less than 180° is called an obtuse angle. Also, recall that a straight angle is equal to 180°. An angle which is greater than 180° but less than 360° is called a reflex angle. Further, two angles whose sum is 90° are called complementary angles, and two angles whose sum is 180° are called supplementary angles.

We have also studied adjacent angles in our earlier classes (see Figure 6.2).

Figure 6.2 : Adjacent angles

Two angles are adjacent, if they have a common vertex, a common arm and their non-common arms are on different sides of the common arm. In Figure 6.2, ∠ABD and ∠DBC are adjacent angles. Ray BD is their common arm and point B is their common vertex. Ray BA and Ray BC are non-common arms. Moreover, when two angles are adjacent, then their sum is always equal to the angle formed by the two noncommon arms. So, we can write

\[∠ABC\ =\ ∠ABD\ +\ ∠DBC.\]

Note that ∠ABC and ∠ABD are not adjacent angles. Why? Because their noncommon arms BD and BC lie on the same side of the common arm BA.

If the non-common arms BA and BC in Figure 6.2, form a line then it will look like Figure 6.3. In this case, ∠ABD and ∠DBC are called linear pair of angles.

Figure 6.3 : Linear pair of angles

We may also recall the vertically opposite angles formed when two lines, say AB and CD, intersect each other, say at the Point O (see Figure 6.4). There are two pairs of vertically opposite angles.

Figure 6.4 : Vertically opposite angles

One pair is \(∠AOD,\ ∠BOC.\) Can we find the other pair?