6. Lines and Angles

Assam board Class 9 Chapter 6. Lines and Angles

Chapter 6. Lines and Angles

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Important Note :

1. If two lines intersect each other, then the vertically opposite angles are equal.
2. If a transversal intersects two parallel lines, then
(i) each pair of corresponding angles is equal,
(ii) each pair of alternate interior angles is equal,
(iii) each pair of interior angles on the same side of the transversal is supplementary.
3. If a transversal intersects two lines such that, either
(i) any one pair of corresponding angles is equal, or
(ii) any one pair of alternate interior angles is equal, or
(iii) any one pair of interior angles on the same side of the transversal is supplementary, then the lines are parallel.
4. Lines which are parallel to a given line are parallel to each other.
5. The sum of the three angles of a triangle is 180°.
6. If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.

EXERCISE 6.1

1. In Fig. 6.13 , lines AB and CD intersect at O . If

and

, find

and reflex

Solution: Given, and .

Since , is a straight line .

and is a straight line .

Reflex

2. In Fig. 6.14 , lines XY and MN intersect at O . If

and

, find

Solution: Given, and .

Let and .

Since, is a straight line .

and

and is a straight line .

Therefore, the value of is 126° .

3. In Fig. 6.15,

, then prove that

Solution: Given , , then we prove that .

Proof : Since, ray stands on line .

………. (i)

Ray stands on line .

…………. (ii)

From (i) and (ii) , we get

But,

Proved .

4. In Fig. 6.16 , If

, then prove that AOB is a line .

Solution : Given, , then we that prove that is a line .

Ray stands on line .

…….(i) [Linear pair of angles]

Ray stands on line .

……(ii) [Linear pair of angles]

Adding (i) and (ii) , we get

[Linear pair of angles]

Therefore, is a line . Proved .

5. In Fig. 6.17 , POQ is a line . Ray OR is perpendicular to line PQ . OS is another ray lying between rays OP and OR . Prove that

Solution: Given , POQ is a line . Ray OR is perpendicular to line PQ . OS is another ray lying between rays OP and OR .

To Prove :

Proof : Since, ray OR is perpendicular to line PQ .

i.e.,

……….. (i)

And

[Add both sides ]

……….. (ii)

From (i) and (ii) , we get

Proved .

6. It is given that

and XY is produced to point P . Draw a figure from the given information . If ray YQ bisects

and find

and reflex

Solution : Given,

Ray YQ bisects , then .

Let

Since, is a straight line .

And

EXERCISE 6.2

1. In Fig. , find the values of and and then show that .

Solution: Since, [vertically opposite angle]

Again, [Linear pair of angles]

But [Alternative interior angle]

Proved.

2. In Fig. ,if , and find .

Solution: Given,

Let, and

Since, and is transversal .

Again, and is transversal .

And

Therefore, the value is 126° .

3. In Fig .6.30, if and , find and .

Solution: Given,

Since, and is a transversal .

[alternative interior angle]

Again,

But

And [Linear pair of angles]

4. In Fig. 6.31, if and find

Solution: Given, and

We draw a line parallel to ST through point R .

i.e., .

Since, and is a transversal

Again, and also , then

[Alternative interior angle ]

5. In Fig . 6.32,if and find and.

Solution: Given, and

Since, and PQ is transversal .

[Alternative interior angle]

And and PR is transversal .

[Alternative interior angle]

6. In Fig .6.33, and are two mirrors placed parallel to each other .An incident ray strikes the mirror at , the reflected ray moves along the path and strikes the mirror at and again reflects back along . prove that .

Solution: Given, and are two mirrors placed parallel to each other .An incident ray strikes the mirror at , the reflected ray moves along the path and strikes the mirror at and again reflects back along .

To prove that : .

Construction : We draw and .

Proof: We know that , the incidence angle and the reflection angle are equal .

So, and

Since, and and also given .

So,

Again, and BC is a transversal .

So, [Alternative interior angle]

Again,

[Alternative interior angle]

So, and is a transversal .

proved.

EXERCISE 6.3

1. In Fig. 6.39, sides and of are produced to points and respectively . If and find

Solution: Given, and

Ray PQ stand on the line TR .

We know that, an exterior angle of a triangle is equal to the sum of the two interior opposite angles.

2.In Fig .6.40 , If and are the bisectors of and respectively of find and .

Solution: Given, and .

In , we have

Since, and are the bisectors of and respectively.

In , we have

Therefore, and .

3. In Fig .6.41 ,if and , find .

Solution: Given, and

and is a transversal .

So,

And

In , we have

4. In Fig .6.42 , if lines and intersect at point ,such that and find

Solution: Given, , and

In , we have

Again, [Vertically opposite angle]

In , we have

5. In Fig. 6.43 , if ,and then find the values of and .

Solution: Given, and .

Since,

So,

We know that, an exterior angle of a triangle is equal to the sum of the two interior opposite angles.

Again, and is a transversal .

[alternative interior angle]

In , we have

Therefore, and

6. In Fig . 6.44, the sides of is produced to a point .If the bisectors of and meet at point , then prove that

Solution: Given, the sides of is produced to a point .If the bisectors of and meet at point .

To prove :

Proof: Since, and are the bisectors of and respectively.

So,

and

We know that, an exterior angle of a triangle is equal to the sum of the two interior opposite angles.

In , we have

From (i) and (ii) , we get

Proved.