`color{red} ♦` Introduction

`color{red} ♦`Basic Terms and Definitions

`color{red} ♦`Intersecting Lines and Non-intersecting Lines

`color{red} ♦`Pairs of Angles

`color{red} ♦`Basic Terms and Definitions

`color{red} ♦`Intersecting Lines and Non-intersecting Lines

`color{red} ♦`Pairs of Angles

As we know 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.

Here,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.

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 multistory building, she has to draw intersecting lines and parallel lines at different angles.

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 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.

Here,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.

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 multistory building, she has to draw intersecting lines and parallel lines at different angles.

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 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.

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 `bar(AB)` , and its length is denoted by AB. The ray AB is denoted by `bar(AB)` , and a line is denoted by `bar(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).

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.

You have also studied about adjacent angles in the earlier classes (see Fig. 6.2). 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 Fig. 6.2, `angle ABD` and `angle 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

`angle ABC = angle ABD + angle DBC`.

Note that `angle ABC` and `angle 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 Fig. 6.2, form a line then it will look like Fig. 6.3. In this case, `angle ABD` and `angle DBC` are called linear pair of angles.

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

One pair is `angle AOD` and `angle BOC`. Can you find the other pair?

`"Note"` that the line segment AB is denoted by `bar(AB)` , and its length is denoted by AB. The ray AB is denoted by `bar(AB)` , and a line is denoted by `bar(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).

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.

You have also studied about adjacent angles in the earlier classes (see Fig. 6.2). 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 Fig. 6.2, `angle ABD` and `angle 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

`angle ABC = angle ABD + angle DBC`.

Note that `angle ABC` and `angle 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 Fig. 6.2, form a line then it will look like Fig. 6.3. In this case, `angle ABD` and `angle DBC` are called linear pair of angles.

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

One pair is `angle AOD` and `angle BOC`. Can you find the other pair?

Draw two different lines PQ and RS on a paper. You will see that you can draw them in two different ways as shown in Fig. 6.5 (i) and Fig. 6.5 (ii).

Recall the notion of a line, that it extends indefinitely in both directions. Lines PQ and RS in Fig. 6.5 (i) are intersecting lines and in Fig. 6.5 (ii) are parallel lines.

Note that the lengths of the common perpendiculars at different points on these parallel lines is the same. This equal length is called the distance between two parallel lines.

Recall the notion of a line, that it extends indefinitely in both directions. Lines PQ and RS in Fig. 6.5 (i) are intersecting lines and in Fig. 6.5 (ii) are parallel lines.

Note that the lengths of the common perpendiculars at different points on these parallel lines is the same. This equal length is called the distance between two parallel lines.

As we've learnt the definitions of some of the pairs of angles such as complementary angles, supplementary angles, adjacent angles, linear pair of angles, etc.

Now, let us find out the relation between the angles formed when a ray stands on a line. Draw a figure in which a ray stands on a line as shown in Fig. 6.6.

Name the line as AB and the ray as OC. What are the angles formed at the point O? They are ` angle AOC, angle BOC` and `angle AOB`. Can we write `angle AOC + angle BOC = angle AOB` ? (1) Yes! (Why? Refer to adjacent angles in Section 6.2 ) What is the measure of `angle AOB` ? It is `180°`.

From (1) and (2), can you say that `angle AOC + angle BOC = 180°` ? Yes ! (Why ?) From the above discussion, we can state the following Axiom :

`color {blue} text(Axiom 6.1 : If a ray stands on a line, then the sum of two adjacent angles so formed is 180°)`.

Recall that when the sum of two adjacent angles is `180°`, then they are called a linear pair of angles.

In Axiom 6.1, it is given that ‘a ray stands on a line’. From this ‘given’, we have concluded that ‘the sum of two adjacent angles so formed is `180°’.` Can we write Axiom 6.1 the other way? That is, take the ‘conclusion’ of Axiom 6.1 as ‘given’ and the ‘given’ as the ‘conclusion’. So it becomes:

(A) If the sum of two adjacent angles is `180°`, then a ray stands on a line (that is, the non-common arms form a line).

Now you see that the Axiom 6.1 and statement (A) are in a sense the reverse of each others. We call each as converse of the other. We do not know whether the statement (A) is true or not.

Let us check. Draw adjacent angles of different measures as shown in Fig. 6.7. Keep the ruler along one of the non-common arms in each case. Does the other non-common arm also lie along the ruler?

You will find that only in Fig. 6.7 (iii), both the non-common arms lie along the ruler, that is, points A, O and B lie on the same line and ray OC stands on it. Also see that `angle AOC + angle COB = 125° + 55° = 180°`. From this, you may conclude that statement (A) is true. So, you can state in the form of an axiom as follows:

Axiom 6.2 : If the sum of two adjacent angles is `180°`, then the non-common arms of the angles form a line.

For obvious reasons, the two axioms above together is called the Linear Pair Axiom.

Let us now examine the case when two lines intersect each other.

Recall, from earlier classes, that when two lines intersect, the vertically opposite angles are equal. Let us prove this result now. See Appendix 1 for the ingredients of a proof, and keep those in mind while studying the proof given below.

Theorem 6.1 : If two lines intersect each other, then the vertically opposite angles are equal.

Proof : In the statement above, it is given that ‘two lines intersect each other’. So, let AB and CD be two lines intersecting at O as shown in Fig. 6.8. They lead to two pairs of vertically opposite angles, namely,

(i) `angle AOC` and `angle BOD ` (ii) `angle AOD` and

`angle BOC.`

We need to prove that `angle AOC = angle BOD`

and `angle AOD = angle BOC`.

Now, ray `OA` stands on line `CD`.

Therefore, `angle AOC + angle AOD = 180°` (Linear pair axiom) (1)

Can we write `angle AOD + angle BOD = 180°`? Yes! (Why?) (2)

From (1) and (2), we can write

`angle AOC + angle AOD = angle AOD + angle BOD`

This implies that `angle AOC = angle BOD` (Refer Section 5.2, Axiom 3)

Similarly, it can be proved that `angle AOD = angle BOC`

Now, let us do some examples based on Linear Pair Axiom and Theorem 6.1.

Now, let us find out the relation between the angles formed when a ray stands on a line. Draw a figure in which a ray stands on a line as shown in Fig. 6.6.

Name the line as AB and the ray as OC. What are the angles formed at the point O? They are ` angle AOC, angle BOC` and `angle AOB`. Can we write `angle AOC + angle BOC = angle AOB` ? (1) Yes! (Why? Refer to adjacent angles in Section 6.2 ) What is the measure of `angle AOB` ? It is `180°`.

From (1) and (2), can you say that `angle AOC + angle BOC = 180°` ? Yes ! (Why ?) From the above discussion, we can state the following Axiom :

`color {blue} text(Axiom 6.1 : If a ray stands on a line, then the sum of two adjacent angles so formed is 180°)`.

Recall that when the sum of two adjacent angles is `180°`, then they are called a linear pair of angles.

In Axiom 6.1, it is given that ‘a ray stands on a line’. From this ‘given’, we have concluded that ‘the sum of two adjacent angles so formed is `180°’.` Can we write Axiom 6.1 the other way? That is, take the ‘conclusion’ of Axiom 6.1 as ‘given’ and the ‘given’ as the ‘conclusion’. So it becomes:

(A) If the sum of two adjacent angles is `180°`, then a ray stands on a line (that is, the non-common arms form a line).

Now you see that the Axiom 6.1 and statement (A) are in a sense the reverse of each others. We call each as converse of the other. We do not know whether the statement (A) is true or not.

Let us check. Draw adjacent angles of different measures as shown in Fig. 6.7. Keep the ruler along one of the non-common arms in each case. Does the other non-common arm also lie along the ruler?

You will find that only in Fig. 6.7 (iii), both the non-common arms lie along the ruler, that is, points A, O and B lie on the same line and ray OC stands on it. Also see that `angle AOC + angle COB = 125° + 55° = 180°`. From this, you may conclude that statement (A) is true. So, you can state in the form of an axiom as follows:

Axiom 6.2 : If the sum of two adjacent angles is `180°`, then the non-common arms of the angles form a line.

For obvious reasons, the two axioms above together is called the Linear Pair Axiom.

Let us now examine the case when two lines intersect each other.

Recall, from earlier classes, that when two lines intersect, the vertically opposite angles are equal. Let us prove this result now. See Appendix 1 for the ingredients of a proof, and keep those in mind while studying the proof given below.

Theorem 6.1 : If two lines intersect each other, then the vertically opposite angles are equal.

Proof : In the statement above, it is given that ‘two lines intersect each other’. So, let AB and CD be two lines intersecting at O as shown in Fig. 6.8. They lead to two pairs of vertically opposite angles, namely,

(i) `angle AOC` and `angle BOD ` (ii) `angle AOD` and

`angle BOC.`

We need to prove that `angle AOC = angle BOD`

and `angle AOD = angle BOC`.

Now, ray `OA` stands on line `CD`.

Therefore, `angle AOC + angle AOD = 180°` (Linear pair axiom) (1)

Can we write `angle AOD + angle BOD = 180°`? Yes! (Why?) (2)

From (1) and (2), we can write

`angle AOC + angle AOD = angle AOD + angle BOD`

This implies that `angle AOC = angle BOD` (Refer Section 5.2, Axiom 3)

Similarly, it can be proved that `angle AOD = angle BOC`

Now, let us do some examples based on Linear Pair Axiom and Theorem 6.1.

Q 3200367218

In Fig. 6.9, lines `PQ` and `RS` intersect each other at point `O`. If `angle POR : angle ROQ = 5 : 7`, find all the angles.

Class 9 Chapter 6 Example 1

Class 9 Chapter 6 Example 1

`angle POR + angle ROQ = 180°`

(Linear pair of angles)

But `angle POR : angle ROQ = 5 : 7`

(Given)

Therefore, `angle POR = 5/(12) xx 180° = 75°`

Similarly, `angle ROQ = 7/(12) xx 180° = 105°`

Now, `angle POS = angle ROQ = 105°` (Vertically opposite angles)

and `angle SOQ = angle POR = 75°`

Q 3210367219

In Fig. 6.10, ray `OS` stands on a line `angle POQ.` Ray `OR` and ray `OT` are angle bisectors of `angle POS` and ` angle SOQ,` respectively. If `angle POS = x`, find `angle ROT`.

Class 9 Chapter 6 Example 2

Class 9 Chapter 6 Example 2

Ray OS stands on the line `POQ`.

Therefore, `angle POS + angle SOQ = 180°`

But, ` angle POS = x`

Therefore, `x + angle SOQ = 180°`

So, `angle SOQ = 180° – x`

Now, ray `OR` bisects `angle POS`, therefore,

` angle ROS = 1/2 xx angle POS`

` = 1/2 xx x = x/2 `

`angle SOT = 1/2 xx angle SOQ`

` = 1/2 xx ( 180^o - x)`

` = 90^0 - x/2`

Now, ` angle ROT = angle ROS + angle SOT`

` = x/2 + 90^0 - x/2`

` = 90^o`

Q 3200467318

In Fig. 6.11, `OP, OQ, OR` and `OS` are four rays. Prove that `angle POQ + angle QOR + angle SOR +` `angle POS = 360°`.

Class 9 Chapter 6 Example 3

Class 9 Chapter 6 Example 3

In Fig. 6.11, you need to produce any of

the rays `OP, OQ, OR` or `OS` backwards to a point.

Let us produce ray `OQ` backwards to a point ` T` so

that `TOQ` is a line (see Fig. 6.12).

Now, ray `OP` stands on line `TOQ`.

Therefore, `angle TOP + angle POQ = 180°` (1)

(Linear pair axiom)

Similarly, ray `OS` stands on line `TOQ`.

Therefore, `angle TOS + angle SOQ = 180°` (2)

But `angle SOQ = angle SOR + angle QOR`

So, (2) becomes

`angle TOS + angle SOR + angle QOR = 180°` (3)

Now, adding (1) and (3), you get

`angle TOP + angle POQ + angle TOS + angle SOR + angle QOR = 360°` (4)

But `angle TOP + angle TOS = angle POS`

Therefore, (4) becomes

`angle POQ + angle QOR + angle SOR + angle POS = 360°`