# Important Questions for CBSE Class 9 Maths Chapter 6 - Lines and Angles

## CBSE Class 9 Maths Chapter-6 Important Questions - Free PDF Download

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1 Marks Questions

1. Measurement of reflex angle is

(i)

(ii) between

(iii) between

(iv) between

Ans. (iv) between

2. The sum of angle of a triangle is

(i)

(ii)

(iii)

(iv) none of these

Ans. (iii)

3. In fig if x= then y=

(i)

(ii)

(iii)

(iv)

Ans. (iii)

4. If two lines intersect each other then

(i) vertically opposite angles are equal

(ii) corresponding angle are equal

(iii) alternate interior angle are equal

(iv) none of these

Ans. (i) vertically opposite angles are equal

5. The measure of Complementary angle of is

(a)

(b)

(c)

(d) none of there

Ans. (c)

6. If two angles of a triangle is 30o and 45o what is measure of third angle

(a)

(b)

(c)

(d)

Ans. (d)

7. The measurement of Complete angle is

(a)

(b)

(c)

(d)

Ans. (d)

8. The measurement of sum of linear pair is

(a)

(b)

(c)

(d)

Ans. (a)

9. The difference of two complementary angles is. The angles are

(a)

(b)

(c)

(d)

Ans. (c)

10. Given two distinct points P and Q in the interior of , then will be

(a) in the interior of

(b) in the interior of

(c) on the

(d) on the both sides of

Ans. (c) on the

11. The complement of is

(a) -

(b)

(c)

(d)

Ans. (d)

12. The number of angles formed by a transversal with a pair of lines is

(a) 6

(b) 3

(c) 8

(d) 4

Ans. (c) 8

13. In fig L1L2 And the measure of 2 is.

(A)

(B)

(C)

(D)

Ans. (B)

14. In fig x= the value of Y is

(A)

(B)

(C)

(D)

Ans. (B)

15. Which of the following pairs of angles are complementary angle?

(A)

(B)

(C)

(D)

Ans. (A)

16. In fig the measure of 1 is.

(A)

(B)

(C)

(D)

Ans. (C)

17. In figure the measure of is

(a)

(b)

(c)

(d)

Ans. (a)

18. The correct statement is-

(a) A line segment has one end point only.

(b) The ray AB is the same as the ray BA.

(c) Three points are collinear if all of them lie on a line.

(d) Two lines are coincident if they have only one point in common.

Ans. (c) Three points are collinear if all of them lie on a line.

19. One angle is five times its supplement. The angles are-

(a)

(b)

(c)

(d)

Ans. (b)

20. In figure if and

(a)

(b)

(c)

(d)

Ans. (b)

2 Marks Questions

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

Ans. We are given thatand.

We need to find.

From the given figure, we can conclude thatform a linear pair.

We know that sum of the angles of a linear pair is.

(Vertically opposite angles), or

But, we are given that

Therefore, we can conclude thatand.

2. In Fig. 6.14, lines XY and MN intersect at O. If  and a:b= 2 : 3, find c.

Ans. We are given thatand.

We need find the value of c in the given figure.

Let a be equal to 2x and b be equal to 3x.

Therefore

Now[Linear pair]

3. In the given figure,, then prove that.

Ans. We need to prove that.

We are given that.

From the given figure, we can conclude thatform a linear pair.

We know that sum of the angles of a linear pair is.

and(i)

(ii)

From equation (i) and (ii), we can conclude that

Therefore, the desired result is proved.

4. In the given figure, find the values of x and y and then show that AB || CD.

Ans. We need to find the value of x and y in the figure given below and then prove that.

From the figure, we can conclude that (Vertically opposite angles), and

form a pair of linear pair.

We know that the sum of linear pair of angles is.

.

From the figure, we can conclude thatform a pair of alternate interior angles corresponding to the lines AB and CD.

Therefore, we can conclude that.

5. In the given figure, if AB || CD, CD || EF and y: z = 3: 7, find x.

Ans. We are given that,and.

We need to find the value of x in the figure given below.

We know that lines parallel to the same line are also parallel to each other.

We can conclude that.

Let.

We know that angles on same side of a transversal are supplementary.

.

(Alternate interior angles)

Now

Therefore, we can conclude that.

6. In the given figure, if AB || CD, and, find x and y.

Ans. We are given that,and.

We need to find the value of x and y in the figure.

(Alternate interior angles)

(Alternate interior angles)

Therefore, we can conclude that.

7. In the given figure, sides QP and RQ of are produced to points S and T respectively. If  and , find PRQ.

Ans. We are given that.

We need to find the value ofin the figure given below.

From the figure, we can conclude thatform a linear pair.

We know that the sum of angles of a linear pair is.

From the figure, we can conclude that

(Angle sum property)

Therefore, we can conclude that.

8. In the given figure, , . If YO and ZO are the bisectors of XYZ and XZY respectively of, find OZY and YOZ.

Ans. We are given thatand YO and ZO are bisectors of, respectively.

We need to findin the figure.

From the figure, we can conclude that in

(Angle sum property)

We are given that OY and OZ are the bisectors of, respectively.

From the figure, we can conclude that in

(Angle sum property)

Therefore, we can conclude thatand.

9. In the given figure, if AB || DE,  and , find DCE.

Ans. We are given that,.

We need to find the value ofin the figure given below.

From the figure, we can conclude that

(Alternate interior angles)

From the figure, we can conclude that in

(Angle sum property)

Therefore, we can conclude that.

10. In the given figure, if lines PQ and RS intersect at point T, such that ,  and , find SQT.

Ans. We are given thatand .

We need to find the value ofin the figure.

From the figure, we can conclude that in

(Angle sum property)

From the figure, we can conclude that

(Vertically opposite angles)

From the figure, we can conclude that in

(Angle sum property)

Therefore, we can conclude that.

11. In fig lines x y and m n intersect at 0 If

Ans. Given in fig. POY=

a: b: 2: 3

Let a=2x and b =3x

a + b +POY=

2x+3x+=

5x=

5x=

x=

MoN is a line.

b+C=

=

12. In fig find the volume of x and y then Show that ABCD

Ans.

(Vertically opposite angles are equal)

x=y as they are corresponding angles.

13. What value of x would make AOB a line if

Ans.

14. In fig POQ is a line. Ray OR is perpendicular to line PQ. OS is another ray lying between rays OP and OR. Prove that

Ans.

……..(i)

= L.H.S

Hence proved.

15. In fig lines P and R intersected at 0, if x =find x, y and u

Ans.

Vertically opposite angles are equal

(By linear pair)

(vertically opposite angles)

16. The exterior angle of a triangle is 110° and one of the interior opposite angle is 35°. Find the other two angles of the triangle.

Ans. The exterior angle of a triangle is equal to the sum of interior opposite angles.

17. Of the three angles of a triangle, one is twice the smallest and another is three times the smallest. Find the angles.

Ans. Let the smallest angle be

Then other two angles are

[sum of three angle of a triangle is 180°]

6x° =180°

x =

=30°

Therefore, angles are

18. Prove that if one angle of a triangle is equal to the sum of other two angles, the triangle is right angled.

Ans. Given in

To prove:  is right angled.

Proof: ….. (1) [Sum of three angles of a ABC is 180°]

….. (2)

From (1) and (2)

19. In fig. sides QP and RQ of  are produced to points S and T respectively. If.

Ans.

Also [Interior angle theorem]

20. In fig the bisector of intersect each other at point O prove that

Ans. Given A  such that the bisectors of meet at a point O

To Prove

Proof: In

(1)

In

[BO and CO bisects]

[Divide forth side by 2]

Substituting,

21. In the given figure  form a linear pair if a – b =. Find the value of ‘a’ and ‘b’.

Ans. [by line as pair]

22. If ray OC stands on a line AB such that , then show that

Ans.

[By lines pair]

23. In the given figure show that ABEF

Ans.

[Alternate interior angles are equal]

Again

[sum of consecutive interior angle is ]

24. In figure if ABCD,Find x and y.

Ans.

is a transversal

[pair of alternate angles]

Also

25. Prove that if two lines intersect each other then vertically opposite angler are equal.

Ans. Given: AB and CD are two lines intersect each other at O.

To prove: (i)  and (ii)

Proof:

[By linear pair]

[By eq (i) and (ii)]

Similarly,

26.The measure of an angle is twice the measure of supplementary angle. Find measure of angles.

Ans. Let the measure be

Then its supplement is

According to question

The measure of the angles are  and .

27. In fig PQR =PRQ. Then prove that PQS=PRT.

Ans.[By linear pair]

But,

[Give]

28. In the given fig AOC =ACO and BOD = BDO prove that ACDB

Ans.[Give]

But,

[vertically opposite angles]

[By alternate interior angle property]

29. In figure if lines PQ and RS intersect at point T. Such that .

Ans.

[By angle sum property]

[vertically opposite angle]

30. In figure, if

Ans.

[Angle sum property of ]

31. In figure sides QP and RQ of  are produced to points S and T respectively if

Ans.  [By linear pair]

[By angle sum property]

32. In figure lines PQ and RS intersect each other at point O. If. Find all the angles.

Ans. [linear pair of angle]

But, [Give]

Similarly,

Now [vertically opposite angle]

And [vertically app angle]

3 Marks Quetions

1. In Fig. 6.16, if x + y = w + z, then prove that AOB is a line.

Ans . We need to prove that AOB is a line.

We are given that.

We know that the sum of all the angles around a fixed point is.

Thus, we can conclude that

But, (Given).

From the given figure, we can conclude that y and x form a linear pair.

We know that if a ray stands on a straight line, then the sum of the angles of linear pair formed by the ray with respect to the line is.

.

Therefore, we can conclude that AOB is a line.

2. In the given figure, POQ is a line. Ray OR is perpendicular to line PQ. OS is another ray lying between rays OP and OR. Prove that

Ans. We need to prove that.

We are given that OR is perpendicular to PQ, or

From the given figure, we can conclude thatform a linear pair.

We know that sum of the angles of a linear pair is.

, or

.

From the figure, we can conclude that.

, or

.(i)

From the given figure, we can conclude thatform a linear pair.

We know that sum of the angles of a linear pair is.

, or

.(ii)

Substitute (ii) in (i), to get

Therefore, the desired result is proved.

3. It is given thatand XY is produced to point P. Draw a figure from the given information. If ray YQ bisects, find .

Ans . We are given that, XY is produced to P and YQ bisects.

We can conclude the given below figure for the given situation:

We need to find.

From the given figure, we can conclude thatform a linear pair.

We know that sum of the angles of a linear pair is.

.

But.

Ray YQ bisects, or

.

Therefore, we can conclude that.

4. In the given figure, If AB || CD,and, find .

Ans. We are given that,and.

We need to find the value of in the figure given below.

(Alternate angles)

From the given figure, we can conclude thatform a linear pair.

We know that sum of the angles of a linear pair is.

(Alternate interior angles)

Therefore, we can conclude that.

5. In the given figure, PQ and RS are two mirrors placed parallel to each other. An incident ray AB strikes the mirror PQ at B, the reflected ray moves along the path BC and strikes the mirror RS at C and again reflects back along CD. Prove that AB || CD.

Ans. We are given that PQ and RS are two mirrors that are parallel to each other.

We need to prove thatin the figure.

Let us draw lines BX and CY that are parallel to each other, to get

We know that according to the laws of reflection

.

(Alternate interior angles)

We can conclude that.

From the figure, we can conclude that

Therefore, we can conclude that.

From the figure, we can conclude thatform a pair of alternate interior angles corresponding to the lines AB and CD, and transversal BC.

Therefore, we can conclude that.

6. In the given figure, if, PQ || SR,, then find the values of x and y.

Ans. We are given that.

We need to find the values of x and y in the figure.

We know that “If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles.”

From the figure, we can conclude that

, or

From the figure, we can conclude that

(Alternate interior angles)

From the figure, we can conclude that

(Angle sum property)

Therefore, we can conclude that.

7. In the given figure, the side QR of ∆PQR is produced to a point S. If the bisectors of meet at point T, then prove that.

Ans. We need to prove thatin the figure given below.

We know that “If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles.”

From the figure, we can conclude that in,is an exterior angle

…..(i)

From the figure, we can conclude that in,is an exterior angle

We are given that are angle bisectors of

We need to substitute equation (i) in the above equation, to get

Therefore, we can conclude that the desired result is proved.

8. Prove that sum of three angles of a triangle is

Ans.

9. It is given that and X Y is produced to point P, draw a fig from the given information. If ray Y Q bisects ZYP, find XYQ and reflex QYP.

Ans.

10. In fig if PQST, PQR = and RST= find QRS.

Ans.
Through point R Draw line Kleist

(Sum of interior angle on the same side of transversal is )

Similarly

11. The side BC of is produced from ray BD. CE is drawn parallel to AB, show that Also prove that.

Ans. AB / CE and Ac intersect them

(1) [Alternate interior angles]

Also AB/CE and BD intersect them

(2) [Corresponding angles]

Adding eq (1) and eq (2)

Adding on both sides, we get

12. Prove that if a transversal intersect two parallel lines, then each pair of alternate interior angles is equal.

Ans. Given: line ABCD intersected by transversal PQ

To Prove: (i) (ii)

Proof: (i) [Vertically Opposite angle]

(ii) [Corresponding angles]

By (i) and (ii)

Similarly,

Hence Proved.

13. In the given figure ABC is right angled at A. AD is drawn perpendicular to BC. Prove that

Ans.

From (1) and (2)

Hence proved.

14. In ABC meets BC at a point D. Find

Ans.

[Sum of three angle of a ]

Also

15. In figure two straight lines AB and CD intersect at a point O. If . Find the value of x hence find

(a)

(b)

(c)

(d)

Ans.  By linear pair

vertically opposite angles

16. The side BC of a ABC is produced to D. the bisector of A meets BC at L as shown if fig. prove that ABC+ACD=2

Ans.

[Exterior angle property]

[exterior angle property]

…(2)

Subtracting (1) from (2)

17. In fig lines XY and MN intersect at O If POY=and a:b=2:3 find C

Ans. Lines XY and MN intersect at O.

[vertically opposite angle]

But,

Also,

But,

a:b = 2:3 [Given]

From (1) and (2) we get

18. In fig PT is the bisector of QPR in PQR and PSQR, find the value of x

Ans.
[Angle sum property of]

[Exterior angle theorem]

19. The sides BA and DC of a quadrilateral ABCD are produced as shown in fig show that X+Y = a+b

Ans.
Join BD

[exterior angle theorem]

20. In the BO and CO are Bisectors of B and C of ABC, show that BOC=+A.

Ans.

And

…(1)

But,

But,

….(2)

From (1) and (2) we get

…..(3)

But,

[angle of a]

21. In fig two straight lines PQ and RS intersect each other at o, if POT= Find the values of a, b and c

Ans.
PQ intersect RS at O
[vertically opposite angles]
A = 4b ….(1)

Also,

[ is a straight lines]

Using, (1)

Or

Again,
form a linear pair

Using, (2)

Hence,

22. In figure ray OS stands on a line POQ, ray OR and ray OT are angle bisector of

Ans.
Ray OS stands on the line POQ

Now ray OR bisects
Therefore
Similarly,

23. If a transversal intersects two lines such that the bisectors of a pair of corresponding angles are parallel, then prove that the two lines are parallel.

Ans.
Given AD is transversal intersect two lines PQ and RS
To prove PQRS
Proof: BE bisects ABQ
=
Similarity C G bisects BCS

But is the transversal

[by (1) and (2)]
[corresponding angles are equal]

24. In figure the sides QR of is produced to a point S. If the bisectors of and meet at point T. Then prove that

Ans.
Solution,

[By exterior angle theorem]

are bisectors of

[By exterior angle theorem]
By eq (1) and (2) we get

Hence proved.

25. In figure PQ and RS are two mirror placed parallel to each other. An incident ray AB striker the mirror PQ at B, the reflected ray moves along the path BC and strike the mirror RS at C and again reflects back along CD. Prove that ABCD.

Ans.
Solution,

[angle of incident]
[is equal to angle of reflection]

[By corresponding angle property]
[alternate interior angle]
By aq (1), (2) and (3)

[by alternate interior angles]

4 Marks Quetions

1. In the given figure, if PQ || ST, and, find. [Hint: Draw a line parallel to ST through point R.]

Ans. We are given that,and.

We need to find the value of in the figure.

We need to draw a line RX that is parallel to the line ST, to get

Thus, we have.

We know that lines parallel to the same line are also parallel to each other.

We can conclude that.

(Alternate interior angles), or

.

We know that angles on same side of a transversal are supplementary.

From the figure, we can conclude that

Therefore, we can conclude that.

2. In fig the side AB and AC of A B C Are produced to point E And D respectively. If bisector BO And CO of CBE And BCD respectively meet at point O, then prove that BOC =BAC

Ans.

Similarly, ray Co bisects

3. In given fig. AB CD. Determine.

Ans. Through O draw a line parallel to both AB and CD

Clearly

[Alternate interior angles]

4. In fig M and N are two plane mirrors perpendicular to each other; prove that the incident ray CA is parallel to reflected ray BD.

Ans. Draw APM and BQ N

[By angle sum property]

Also

[By sum of interior angles of same side of transversal]

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

Ans. Solution,