Chapter 10 : Lines and Angles - Rd Sharma Solutions for Class 9 Maths CBSE

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Chapter 10 - Lines and Angles Excercise Ex. 10.1

Question 1

Write the complement of each of the following angles:

(i) 20o

(ii) 35o

(iii) 90o

(iv) 77o

(v) 30o

Solution 1

Question 2

Write the supplement of each of the following angles:

(i) 54o

(ii) 132o

(iii) 138o

Solution 2

(i) 54°

Since, the sum of an angle and its supplement is 180°

∴Its supplement will be 180° - 54° = 126°.


(ii) 132°

Since, the sum of an angle and its supplement is 180°

∴Its supplement will be 180° - 132° = 48°.


(iii) 138°

Since, the sum of an angle and its supplement is 180°

∴Its supplement will be 180° - 138° = 42°.

Question 3

If an angle is 28o less than its complement, find its measure.

Solution 3

Question 4

If an angle is 30o more than one half of its complement, find the measure of the angle.

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

If the complement of an angle is equal to the supplement of the thrice of it. Find the measure of angle.

Solution 13

Let the measure of the angle be xo.

Its complement will be (90o - xo) and its supplement will be (180o - xo).

Supplement of thrice of the angle = (180o - 3xo)

According to the given information:

(90o - xo) = (180o - 3xo)

3x - x = 180 - 90

2x = 90

x = 45

Thus, the measure of the angle is 45o.

The measure of the angle is 45o

Question 14

Solution 14

Chapter 10 - Lines and Angles Excercise Ex. 10.2

Question 1

In fig., OA and OB are opposite rays:

(i) If x = 25°, what is the value of y?

(ii) if y = 35°, what is the value of x?

 

Solution 1

 

Question 2

In fig., write all pairs of adjacent angles and all the linear pairs.

 

Solution 2

Question 3

In fig., find x. further find ∠BOC, ∠COD and ∠AOD

 

Solution 3

 

 

 

Question 4

 

 

 

 

 

 

In fig., rays OA, OB, OC, OD and OE have the common end point O. Show that  ∠AOB + ∠BOC + ∠COD + ∠DOE + ∠EOA = 360º

Solution 4

 

Question 5

 

Solution 5

 

Question 6

Solution 6

Question 7

How many pairs of adjacent angles, in all, can you name in fig.

Solution 7

Question 8

In fig., determine the value of x.

Solution 8

Question 9

In fig., AOC is a line, find x.

 

Solution 9

Question 10

In Fig., POS is a line, find x.

Solution 10

Question 11

In fig., ACB is a line such that ∠DCA = 5x and ∠DCB = 4x. Find the values of ∠DCA and∠DCB

Solution 11

Question 12

Give POR = 3x and QOR = 2x + 10, find the value of x for which POQ will be aline.

Solution 12

Question 13

What value of y would make AOB a line in fig., if ∠AOC = 4y and ∠BOC = (6y + 30)?

 

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

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

 

 

 

 

Solution 16

Question 17

In Fig. If a greater than b by one third of a right-angle. find the values of a and b.

Solution 17

Question 18

 

 

Solution 18

Question 19
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
                                   
Solution 19
Given that OR ⊥ PQ  
 
∴ POR = 90o   
 
    ⇒ ∠POS  + ∠SOR = 90o
 
∠ROS = 90o  - ∠POS                ... (1)
 
    ∠QOR = 90o                     (As OR ⊥ PQ) 
 
   ∠QOS - ∠ROS = 90o
 
    ∠ROS = ∠QOS - 90o             ... (2)
 
    On adding equations (1) and (2), we have
 
    2 ∠ROS = ∠QOS - ∠POS

Chapter 10 - Lines and Angles Excercise Ex. 10.3

Question 1

In fig., lines l1 aans l2 intersect at O, forming angles as shown in the figure. If x = 45, find the values of y, z and u.

 

 

Solution 1

Question 2

In fig., three coplanar lines intersect at a point O, forming angles as shown in the figure. Find the values of x, y, z and u.

 

Solution 2

Question 3

In fig. , find the values of x, y and z.

 



Solution 3

Question 4

In Fig., find the value of X.

 

Solution 4

Question 5

Solution 5



Question 6

In fig., rays AB and CD intersect at O.

(i) Determine y when x = 60o

(ii) Determine x when y = 40

Solution 6

Question 7

In fig., lines AB, CD and EF intersect at O. Find the measures of ∠AOC, ∠COF, ∠DOE and ∠BOF.

 

Solution 7

Question 8

Solution 8



Question 9

In fig., lines AB, and CD intersect at O. If ∠AOC + ∠BOE = 70° and ∠BOD = 40°, find ∠BOE and reflex ∠COE.

 

 

Solution 9

Question 10

Which of the following statements are true (T) and which are false (F)?

(i) Angles forming a linear pair are supplementary.

(ii) If two adjacent angles are equal, then each angle measures 90o.

(iii) Angles forming a linear pair can both be acute angles.

(iv) If angles forming a linear pair are equal, then each of these angles is of measure 90o.

Solution 10

(i) True

(ii) False

(iii) False

(iv) True

Question 11

Fill in the blanks so as to make the following statements true:

(i) If one angle of a linear pair is acute, then its other angle will be ________.

(ii) A ray stands on a line, then the sum of the two adjacent angles so formed is _______.

(iii) If the sum of two adjacent angles is 180o, then the ______ arms of the two angles are opposite rays.

Solution 11

(i) obtuse.

(ii) 180o

(iii) uncommon

Question 12

Solution 12



Question 13

Solution 13



Chapter 10 - Lines and Angles Excercise Ex. 10.4

Question 1

In fig., AB ∥ CD and ∠1 and ∠2 are in the ratio 3:2. determine all angles from 1 to 8.

 

 

Solution 1

Question 2

In fig., l, m and n are parallel lines intersected by transversal p at x, y and z respectively. find ∠1, ∠2, ∠3.

 

 

 

 

 

Solution 2

Question 3

In fig., if AB ∥ CD and CD ∥ EF, find ∠ACE.

 

Solution 3

Question 4

In fig., state which lines are parallel and why.

Solution 4

Question 5

In fig. if l ∥ m, n ∥ p and ∠1 = 85°, find ∠2.

 

Solution 5

Question 6

If two straight lines are perpendicular to the same line, prove that they are parallel to each other.

Solution 6



Question 7

Solution 7



Question 8

Solution 8

Let AB and CD be perpendicuar to line MN.


Question 9

In fig., ∠1 = 60° and ∠2 = (2/3)rd of a right angle. prove that l ∥ m

 

 

 


Solution 9

Question 10

In fig., if l ∥ m ∥ n and ∠1 = 60°, find ∠2.


Solution 10

Question 11



Solution 11

Let AB and CD be perpendicuar to line MN.


Question 12

Solution 12



Question 13

Solution 13




Question 14

In fig., p is transversal to lines m and n, ∠2 = 120° and ∠5 = 60°. Prove that m ∥ n. 

 

Solution 14

Question 15

In fig., transceral l intersects two lines m and n, ∠4 = 110° and ∠7 = 65°. is m ∥ n ?

Solution 15

Question 16

Which pair of lines in Fig., are parallel? give reasons.

 
Solution 16

Question 17

Solution 17


Question 18

Solution 18

Question 19

Solution 19

Question 20

In fig., AB ∥ CD ∥ EF and GH ∥ KL. Find ∠HKL

Solution 20

Question 21

In fig., show that AB ∥ EF.

Solution 21

Question 22

In Fig., PQ ∥ AB and PR BC. IF ∠QPR = 102º, determine ∠ABC. Give reasons.

 

 

Solution 22

Question 23

Prove that if the two arms of an angle are perpendicular to the two arms of another angle, then the angles are either equal or supplementary.

Solution 23

Consider the angles AOB and ACB.



Question 24

In fig.,  lines AB and CD are parallel and p is any point as shown in the figure. Show that ∠ABP + ∠CDP = ∠DPB.

Solution 24

Question 25

In fig., AB ∥ CD and P is any point shown in the figure. Prove that:

∠ABP + ∠BPD + ∠CDP = 360°

Solution 25

Question 26

In fig., arms BA and BC of ∠ABC are respectively parallel to arms ED and EF of ∠DEF. Prove that ∠ABC = ∠DEF

 

 
Solution 26

 



Question 27

 

 
Solution 27

Chapter 10 - Lines and Angles Excercise 10.51

Question 1

One angle is equal to three times its supplement. The measure of the angle is

(a) 130°

(b) 135°

(c) 90°

(d) 120°

Solution 1

begin mathsize 11px style Let space the space required space angle space be space straight theta. Then comma space measure space of space its space supplement space equals space 180 degree space minus space straight theta According space to space question comma space we space have straight theta space equals space 3 open parentheses 180 space minus space straight theta close parentheses rightwards double arrow straight theta equals 540 to the power of ring operator space minus space 3 straight theta rightwards double arrow 4 straight theta space equals space 540 to the power of ring operator rightwards double arrow straight theta space equals space 135 to the power of ring operator Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 2

begin mathsize 11px style Two space straight space lines space AB space and space CD space intersect space one space another space at space the space point space straight O. space If space angle space AOC space plus space angle space COB space plus space angle space BOD space equals space 274 to the power of ring operator comma space then space angle space AOD space equals left parenthesis straight a right parenthesis space 86 to the power of ring operator left parenthesis straight b right parenthesis space 90 to the power of ring operator left parenthesis straight c right parenthesis space 94 to the power of ring operator left parenthesis straight d right parenthesis space 137 to the power of ring operator end style

Solution 2

begin mathsize 11px style angle space AOC space plus space angle space COB space plus space angle space BOD space plus space angle space AOD space equals space 360 to the power of ring operator space space space space.... left parenthesis 1 right parenthesis Now comma space angle space AOC space plus space angle space COB space plus space angle space BOD space equals space 274 to the power of ring operator space space space space space space open parentheses Given close parentheses space.... left parenthesis 2 right parenthesis From space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis comma 274 to the power of ring operator plus space angle space AOD space equals space 360 ring operator rightwards double arrow angle space AOD space equals space 360 degree space minus space 274 degree rightwards double arrow angle space AOD space equals space 86 to the power of ring operator Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 3

begin mathsize 12px style Two space straight space lines space AB space and space CD space cut space each space other space at space apostrophe straight O apostrophe. If space angle space BOD space equals space 63 to the power of ring operator comma space then space angle space BOC space equals left parenthesis straight a right parenthesis space 63 to the power of ring operator left parenthesis straight b right parenthesis space 117 to the power of ring operator left parenthesis straight c right parenthesis space 17 to the power of ring operator left parenthesis straight d right parenthesis space 153 to the power of ring operator end style

Solution 3

begin mathsize 11px style angle space BOD space and space angle space BOC space form space straight a space linear space pair. therefore space angle space BOD space plus space angle space BOC space equals space 180 to the power of ring operator rightwards double arrow 63 to the power of ring operator space plus space angle space BOC space equals space 180 to the power of ring operator rightwards double arrow angle space BOC space equals 117 to the power of ring operator Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 4

begin mathsize 11px style Consider space the space following space statements space colon when space two space straight space lines space intersect space colon left parenthesis straight i right parenthesis space adjacent space angles space are space complementary left parenthesis ii right parenthesis space adjacent space angles space are space supplementary left parenthesis iii right parenthesis space opposite space angles space are space equal left parenthesis iv right parenthesis space opposite space angles space are space supplementary Of space these space statements left parenthesis straight a right parenthesis space left parenthesis straight i right parenthesis space and space left parenthesis iii right parenthesis space are space correct left parenthesis straight b right parenthesis space left parenthesis ii right parenthesis space and space left parenthesis iii right parenthesis space are space correct left parenthesis straight c right parenthesis space left parenthesis straight i right parenthesis space and space left parenthesis iv right parenthesis space are space correct left parenthesis straight d right parenthesis space left parenthesis ii right parenthesis space and space left parenthesis iv right parenthesis space are space correct end style

Solution 4

begin mathsize 12px style Let space two space lines space AB space and space CD space intersect space each space other space at space straight O. Now space we space can space see space from space figure space any space two space adjacent space angles comma angle space AOD space & space angle space DOB comma space angle space DOB space & space angle space BOC space etc space are space supplementary space because space their space sum space is space 180 degree. angle space AOD space plus space angle space DOB space equals space 180 to the power of ring operator angle space DOB space plus space angle BOC space equals space 180 to the power of ring operator So space two space adjacent space angles space are space always space supplementary. Now space two space opposite space angle space like space angle space AOC space & space angle space DOB comma space angle space AOD space & space angle space COB are space always space equal space to space each space other space as space they space are space vertically space opposite space angles angle space AOC space equals space angle space DOB angle space AOD space equals space angle space COB Hence space statement space left parenthesis ii right parenthesis space and space left parenthesis iii right parenthesis space are space correct Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 5

begin mathsize 11px style Given space angle POR space equals space 3 straight x space and space angle QOR space equals space 2 straight x space plus space 10 to the power of ring operator space If space POQ space is space straight a space straight space line space then space the space value space of space straight x space is left parenthesis straight a right parenthesis space 30 to the power of ring operator space left parenthesis straight b right parenthesis space 34 to the power of ring operator left parenthesis straight c right parenthesis space 36 to the power of ring operator left parenthesis straight d right parenthesis space None space of space these space end style

Solution 5

begin mathsize 11px style angle POR space equals space 3 straight x space and space angle QOR space equals space 2 straight x space plus 10 to the power of ring operator space From space figure comma space we space can space see space that space angle space POR space & space angle space QOR space are space two space adjacent space angles space and space are space supplement. rightwards double arrow angle POR space plus space angle QOR space equals space 180 to the power of ring operator rightwards double arrow 3 straight x space plus space 2 straight x space plus space 10 to the power of ring operator space equals 180 to the power of ring operator space space rightwards double arrow 5 straight x space equals space 170 to the power of ring operator space rightwards double arrow straight x space equals space 34 to the power of ring operator straight H ence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 6

begin mathsize 11px style In space f ig u r e comma space AOB space is space straight a space straight space line. space If space angle AOC space plus space angle BOD space equals space 85 to the power of ring operator comma space then space angle COD space equals left parenthesis straight a right parenthesis space 85 to the power of ring operator left parenthesis straight b right parenthesis space 90 to the power of ring operator left parenthesis straight c right parenthesis space 95 to the power of ring operator left parenthesis straight d right parenthesis space 100 to the power of ring operator end style

Solution 6

begin mathsize 11px style From space Figure comma space we space can space see space angle AOC space plus space angle COD space plus space angle BOD space equals space 180 to the power of ring operator But comma space angle AOC space plus space angle BOD space equals space 85 to the power of ring operator space space space space space.... left parenthesis Given right parenthesis rightwards double arrow 85 to the power of ring operator space plus space angle COD space equals space 180 to the power of ring operator rightwards double arrow angle COD space equals space 95 to the power of ring operator Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 7

In figure, the value of y is

(a) 20°

(b) 30°

(c) 45°

(d) 60°

Solution 7

begin mathsize 11px style From space figure comma space we space can space see space angle straight x to the power of ring operator equals angle space straight y to the power of ring operator space space space space space left parenthesis vertically space opposite space angles right parenthesis space Also comma space angle 3 straight x degree plus angle straight y degree plus angle 2 straight x degree space equals space 180 degree Now comma space angle straight x degree equals angle straight y to the power of ring operator therefore space angle 3 straight y to the power of ring operator space plus space angle straight y to the power of ring operator space plus space angle 2 straight y to the power of ring operator space equals space 180 to the power of ring operator rightwards double arrow angle 6 straight y to the power of ring operator space equals space 180 to the power of ring operator rightwards double arrow angle straight y to the power of ring operator equals 30 to the power of ring operator Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Chapter 10 - Lines and Angles Excercise 10.52

Question 1

In figure, the value of x is

(a) 12

(b) 15

(c) 20

(d) 30

 

Solution 1

begin mathsize 12px style From space figure comma space we space can space see space that angle BOD space plus space angle AOD space equals space 180 degree angle BOD space equals space 90 degree space left parenthesis Given right parenthesis rightwards double arrow angle AOD space equals space 180 degree space minus space 90 degree space equals space 90 degree Now comma space straight x degree space equals space angle COE space equals space angle FOD space space space space space left parenthesis opposite space angles space are space equal right parenthesis Now comma space angle AOF space plus space angle FOD space equals space 90 degree space equals space angle AOD rightwards double arrow 3 straight x degree space plus space 10 degree space plus space straight x degree space equals space 90 degree rightwards double arrow 4 straight x degree space equals space 80 degree rightwards double arrow straight x degree space equals space 20 degree Hence comma space correct space opiton space is space left parenthesis straight c right parenthesis. end style

Question 2

In figure, which of the following statements must be true?

(i) a + b = d + c     (ii)  a + c + e = 180°          (ii)  b + f = c + e

(a) (i) only

(b) (ii) only

(c) (iii) only

(d) (ii) and (iii) only

 

Solution 2

begin mathsize 12px style From space figure comma space we space can space see space that angle straight a degree space plus space angle straight b degree space plus space angle straight c degree space equals space angle FOC space equals space 180 degree Also comma space angle straight b degree space equals space angle straight e degree space space space space left parenthesis opposite space angles right parenthesis So comma space angle straight a degree space plus space angle straight e degree space plus space angle straight c degree space equals space 180 degree space space space space space space rightwards double arrow space left parenthesis ii right parenthesis space is space correct Now comma space angle FOB space not equal to space angle DOB rightwards double arrow angle straight a degree space plus space angle straight b degree space not equal to space angle straight d degree space plus space angle straight c degree space space rightwards double arrow left parenthesis straight i right parenthesis space is space incorrect Now comma space space angle straight b degree space equals space angle straight e degree space and space angle straight f degree space equals space angle straight c degree space space space left parenthesis opposite space angles space are space equal right parenthesis Thus comma space angle straight b degree space plus space angle straight f degree equals angle straight e degree space plus space angle straight c degree space rightwards double arrow left parenthesis iii right parenthesis space is space also space correct Hence comma space correct space option space is space left parenthesis straight d right parenthesis. end style

Chapter 10 - Lines and Angles Excercise 10.53

Question 1

If two interior angles on the same side of a transversal intersecting two parallel lines are in the ratio 2 : 3, then the measure of the larger angle is

(a) 54°

(b) 120°

(c) 108° 

(d) 136°

Solution 1

begin mathsize 12px style Let space AB space and space CD space are space two space Parallel space lines space and space PQ space is space transverce space to space it. According space to space Question comma fraction numerator angle BRS over denominator angle DSR end fraction equals 2 over 3 rightwards double arrow angle BRS space equals space 2 over 3 angle DSR space space space.... left parenthesis 1 right parenthesis Now comma space angle CSR space equals space angle BRS space space space space left parenthesis Alternate space angles right parenthesis Also comma space angle CSR space plus space angle DSR space equals space 180 degree rightwards double arrow angle BRS space plus space angle DSR space equals space 180 degree rightwards double arrow 2 over 3 angle DSR space plus space angle DSR space equals space 180 degree rightwards double arrow angle DSR space equals fraction numerator begin display style stack up diagonal strike 180 with 36 on top cross times 3 end style over denominator up diagonal strike 5 end fraction equals 108 degree rightwards double arrow angle BRS equals 2 over 3 cross times 108 degree equals 72 degree Thus comma space angle DSR space equals space 108 degree space and space angle BRS space equals space 72 degree rightwards double arrow Larger space angle space is space angle DSR. Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 2

begin mathsize 12px style In space figure comma space AB space parallel to space CD comma space then space the space value space of space straight x space is left parenthesis straight a right parenthesis space 20 to the power of ring operator left parenthesis straight b right parenthesis space space 30 to the power of ring operator left parenthesis straight c right parenthesis space space 45 to the power of ring operator left parenthesis straight d right parenthesis space space 60 to the power of ring operator end style

Solution 2

begin mathsize 12px style From space figure comma angle DPQ space plus space angle straight x degree space equals space 180 degree space space left parenthesis linear space pair right parenthesis space space space.... left parenthesis 1 right parenthesis Also comma space angle DPQ space equals space angle AQP space space space space space left parenthesis interior space opposite space angles right parenthesis rightwards double arrow angle DPQ space equals space 120 degree space plus space straight x space space From space left parenthesis 1 right parenthesis comma space 120 degree space plus space straight x space plus space straight x equals space 180 degree rightwards double arrow 2 straight x space equals space 60 degree rightwards double arrow straight x space equals space 30 degree Hence comma space option space left parenthesis straight b right parenthesis space is space correct. end style

Question 3

begin mathsize 12px style Two space lines space AB space and space CD space intersect space at space straight O. space If space angle AOC space plus space angle COB space plus space angle BOD space equals space 270 to the power of ring operator comma space then space angle AOC space equals left parenthesis straight a right parenthesis space 70 to the power of ring operator left parenthesis straight b right parenthesis space 80 to the power of ring operator left parenthesis straight c right parenthesis space 90 to the power of ring operator left parenthesis straight d right parenthesis space 180 to the power of ring operator end style

Solution 3

begin mathsize 12px style angle AOC space plus space angle COB space plus space angle BOD space equals space 270 degree space space space left parenthesis Given right parenthesis From space figure comma angle AOC space plus space angle COB space plus space angle BOD space plus space angle DOA space equals space 360 degree rightwards double arrow 270 degree space plus space angle DOA space equals space 360 degree rightwards double arrow angle DOA space equals space 360 degree space minus space 270 degree space equals space 90 degree Now space angle DOA space plus space angle AOC space equals space 180 to the power of ring operator rightwards double arrow angle AOC space equals space 180 degree space minus space 90 degree space equals space 90 degree Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 4

begin mathsize 12px style In space figure comma space PQ space parallel to space RS comma space angle AEF space equals space 95 to the power of ring operator comma space angle BHS space equals 110 to the power of ring operator space and space angle space ABC space equals space straight x to the power of ring operator. space Then space the space value space of space straight x space is left parenthesis straight a right parenthesis space 15 to the power of ring operator left parenthesis straight b right parenthesis space 25 to the power of ring operator left parenthesis straight c right parenthesis space 70 to the power of ring operator left parenthesis straight d right parenthesis space 35 to the power of ring operator space end style

Solution 4

begin mathsize 12px style From space figure comma angle AEF space equals space angle EGH space space space space space space left parenthesis Corresponding space angles right parenthesis rightwards double arrow angle EGH equals angle AEF space equals 95 degree space space Also comma space angle BGH space plus space angle EGH space equals space 180 degree rightwards double arrow angle BGH space equals space 180 degree space minus space angle EGH space equals space 180 degree space minus space 95 degree space equals 85 degree angle BHS space equals 110 degree Now comma space angle BHG space plus space angle space BHS space equals 180 degree rightwards double arrow angle BHG space equals space 180 degree space minus space angle BHS equals space 180 degree space minus space 110 degree equals 70 degree space Now comma space in space triangle BHG angle BGH space plus space angle BHG space plus space straight x space equals space 180 degree space space space left parenthesis Sum space of space all space angles space of space straight a space triangle space is space 180 degree right parenthesis rightwards double arrow 8 5 degree plus 70 degree space plus space straight x degree space equals space 180 degree rightwards double arrow straight x degree space equals space 180 degree space minus space 155 degree space equals 25 degree Hence comma space correct space option space is space left parenthesis straight b right parenthesis. space end style

Question 5

begin mathsize 12px style In space figure comma space if space straight l subscript 1 space parallel to space straight l subscript 2 space comma space what space is space the space value space of space straight x ? left parenthesis straight a right parenthesis thin space 90 to the power of ring operator left parenthesis straight b right parenthesis space 85 to the power of ring operator left parenthesis straight c right parenthesis space 75 to the power of ring operator left parenthesis straight d right parenthesis space 70 to the power of ring operator end style

Solution 5

begin mathsize 12px style From space figure comma space angle ERC space equals space angle RPA space space space space space space left parenthesis Correspondence space angles space are space equal right parenthesis rightwards double arrow angle ERC space equals space 37 degree space equals space angle RPA Also comma space angle RPA space equals space angle BPF space space space space space space space space space left parenthesis opposite space angles right parenthesis rightwards double arrow angle RPA space equals space 37 degree space equals space angle BPF Now comma space angle QPB space plus space angle BPF space plus space angle FPG space equals space 180 degree rightwards double arrow straight x degree plus space 37 degree plus space 58 degree space equals space 180 degree rightwards double arrow straight x degree space equals space 85 degree Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Chapter 10 - Lines and Angles Excercise 10.54

Question 1

begin mathsize 12px style In space figure comma space if space straight l subscript 1 space parallel to space straight l subscript 2 comma space what space is space straight x space plus space straight y space in space terms space of space straight w space and space straight z ? left parenthesis straight a right parenthesis space 180 space minus straight w space plus space straight z left parenthesis straight b right parenthesis space 180 space plus space straight w space minus space straight z left parenthesis straight c right parenthesis space 180 space minus space straight w space minus space straight z left parenthesis straight d right parenthesis space 180 space plus space straight w space plus space straight z end style

Solution 1

begin mathsize 12px style Let space angle space supplement space of space straight w degree space be space straight a degree. rightwards double arrow straight a degree space equals space 180 degree space minus space straight w degree Now comma space straight a degree space equals space straight x degree space space space space space space space space space open parentheses Alternate space opposite space angles close parentheses rightwards double arrow straight x degree space equals space 180 degree space minus space straight w degree space space.... left parenthesis 1 right parenthesis Now comma space straight y degree space equals space straight z degree space space space.... left parenthesis 2 right parenthesis space space open parentheses Alternate space space angles close parentheses Adding space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis comma space we space get straight x degree space plus space straight y degree space equals space 180 degree space minus space straight w degree space plus space straight z degree Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 2

begin mathsize 12px style In space figure comma space if space straight l subscript 1 space parallel to space straight l subscript 2 comma space what space is space the space value space of space straight y ? left parenthesis straight a right parenthesis space 100 left parenthesis straight b right parenthesis space 120 left parenthesis straight c right parenthesis space 135 left parenthesis straight d right parenthesis space 150 end style

Solution 2

begin mathsize 12px style Let space angle space supplement space of space 3 straight x degree space be space straight z degree rightwards double arrow straight z degree space equals space 180 degree space minus space 3 straight x degree rightwards double arrow angle AHF space plus space angle FHB space equals space 180 degree rightwards double arrow straight z degree space plus space 3 straight x degree space equals space 180 degree rightwards double arrow straight z degree space equals space 180 degree space minus space 3 straight x degree Now comma space straight x degree space plus space straight y degree space equals space 180 degree Also comma space straight x degree space equals space straight z degree space space space space space left parenthesis correspondence space angles right parenthesis rightwards double arrow space straight x degree space equals space 180 degree space minus space 3 straight x degree rightwards double arrow space 4 straight x degree space equals space 180 degree rightwards double arrow space straight x degree space equals space 45 degree straight x degree space plus space straight y degree space equals space 180 degree rightwards double arrow straight y degree space equals space 180 degree space minus space straight x degree space equals space 180 degree space minus space 45 degree space equals space 135 degree Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 3

begin mathsize 12px style In space f i g u r e comma space if space straight l subscript 1 space parallel to space straight l subscript 2 space and space straight l subscript 3 space parallel to space straight l subscript 4 space comma space What space is space straight y space in space terms space of space straight x ? left parenthesis straight a right parenthesis space 90 space plus space straight x left parenthesis straight b right parenthesis space 90 space plus space 2 straight x left parenthesis straight c right parenthesis space 90 minus straight x over 2 left parenthesis straight d right parenthesis space 90 space minus space 2 straight x end style

Solution 3

begin mathsize 12px style Fom space figure comma angle EPR space equals space angle PQS space space space space space open parentheses Correspondence space angles space are space equal close parentheses rightwards double arrow angle PQS equals space straight x degree straight A lso comma space angle PQS space equals angle RSD space open parentheses Correspondence space angles space are space equal close parentheses rightwards double arrow angle RSD space equals space straight x degree Now comma space angle RSD space plus space straight y degree space plus space straight y degree space equals space 180 degree rightwards double arrow straight x degree plus space 2 straight y degree space equals 180 degree rightwards double arrow straight y degree equals fraction numerator 180 degree minus straight x degree over denominator 2 end fraction rightwards double arrow straight y degree equals 90 degree minus fraction numerator straight x degree over denominator 2 end fraction Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Chapter 10 - Lines and Angles Excercise 10.55

Question 1

begin mathsize 12px style In space figure comma space if space straight l space parallel to space straight m comma space What space is space the space value space of space straight x ? left parenthesis straight a right parenthesis space 60 to the power of ring operator left parenthesis straight b right parenthesis space 50 to the power of ring operator left parenthesis straight c right parenthesis space 45 to the power of ring operator left parenthesis straight d right parenthesis space 30 to the power of ring operator end style

Solution 1

begin mathsize 12px style 3 straight y degree space equals space 2 straight y degree space plus space 25 degree space space space space space space left parenthesis Alternate space angles right parenthesis rightwards double arrow straight y degree space equals space 25 degree space Now comma space straight x degree space plus space 15 degree space equals space 2 straight y degree space plus space 25 degree space space space space space space left parenthesis opposite space angles right parenthesis rightwards double arrow straight x space equals space 2 straight y degree space plus space 25 degree minus 15 degree rightwards double arrow straight x equals 2 straight y degree space plus space 10 degree rightwards double arrow straight x equals 2 cross times space 25 degree plus 10 degree rightwards double arrow straight x space equals 60 degree Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 2

begin mathsize 12px style In space figure comma space if space AB space parallel to space HF space and space DE space parallel to space FG comma space then space the space measure space of space angle FDE space is left parenthesis straight a right parenthesis space 108 to the power of ring operator left parenthesis straight b right parenthesis space 80 to the power of ring operator left parenthesis straight c right parenthesis space 100 to the power of ring operator left parenthesis straight d right parenthesis space 90 to the power of ring operator end style

Solution 2

begin mathsize 12px style AB space parallel to space HF space AND space angle CFH equals 28 degree space space space space space space space left parenthesis given right parenthesis angle CFH space equals space angle FDA space space space space open parentheses Correspondence space Angles space are space equal close parentheses rightwards double arrow angle FDA space equals space space 28 degree Now comma space angle FDA space plus space angle FDE space plus space angle EDB space equals space 180 degree rightwards double arrow 28 degree plus angle FDE plus 72 degree equals 180 degree rightwards double arrow angle FDE space equals 80 degree Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 3

In figure, if lines l and m are parallel, then x = 

(a) 20°

(b) 45°

(c) 65°

(d) 85°

Solution 3

begin mathsize 12px style From space figure comma angle ABD space equals space angle CDF space space left parenthesis Correspondence space Angles right parenthesis rightwards double arrow angle CDF equals space 65 degree Now comma space angle FDE space equals space 180 degree space minus space angle CDF space equals 180 degree space minus space 65 degree rightwards double arrow angle FDE space equals space 115 degree In space triangle EDF comma angle FDE space plus space angle DEF space plus space angle EFD space equals space 180 degree rightwards double arrow 115 degree space plus space straight x space plus space 20 degree space equals space 180 degree space open parentheses Sum space of space all space interior space angles space of space straight a space triangle space as space 180 degree close parentheses rightwards double arrow straight x space equals space 180 degree space minus space 20 degree space minus 115 degree equals space 45 degree Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 4

begin mathsize 12px style In space figure comma space if space AB space parallel to space CD comma space then space straight x space equals left parenthesis straight a right parenthesis space 100 to the power of ring operator left parenthesis straight b right parenthesis space 105 to the power of ring operator left parenthesis straight c right parenthesis space 110 to the power of ring operator left parenthesis straight d right parenthesis space 115 to the power of ring operator end style

Solution 4

begin mathsize 12px style Extending space line space BA space and space CP space to space meet space at space Point space straight E. angle APE space equals space 180 degree space minus space 148 degree space equals space 32 degree angle EAP space equals space 180 degree space minus space 132 degree space equals 48 degree angle AEP space equals straight x degree space space space space open curly brackets left parenthesis Correspondence space angles right parenthesis space because space AB space parallel to space CD space cut space by space transverse space EC close curly brackets Now comma space in space triangle APE angle APE space plus space angle EAP space plus space angle AEP space equals space 180 degree rightwards double arrow 32 degree space plus space 48 degree space plus space straight x degree space equals space 180 degree rightwards double arrow straight x space equals space 100 degree Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Chapter 10 - Lines and Angles Excercise 10.56

Question 1

In figure, if lines l and m are parallel lines, then x = 

(a) 70°

(b) 100°

(c) 40°

(d) 30°

Solution 1

begin mathsize 12px style From space figure comma angle ABC space equals space angle DCE space space space space space..... left parenthesis 1 right parenthesis space space space space open parentheses Corresponding space angles close parentheses angle ECF space equals space 180 degree space minus space angle DCE space space space left parenthesis Supplementry right parenthesis space space space space space space space space space space space space space equals space 180 degree space minus space angle ABC space space space space left square bracket From space left parenthesis 1 right parenthesis right square bracket space space space space space space space space space space space space space space equals space 180 degree space minus space 70 degree rightwards double arrow angle ECF space equals space 110 degree Now comma space in space triangle CEF angle ECF space plus space angle CFE space plus space angle FEC space equals space 180 degree rightwards double arrow 110 degree space plus space straight x space plus space 30 degree space equals space 180 degree rightwards double arrow straight x space equals space 40 degree Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 2

begin mathsize 12px style In space figure comma space if space straight l space parallel to space straight m comma space then space straight x space equals left parenthesis straight a right parenthesis space 105 degree left parenthesis straight b right parenthesis space 65 degree left parenthesis straight c right parenthesis space 40 degree left parenthesis straight d right parenthesis space 25 degree end style

Solution 2

begin mathsize 12px style From space figure comma angle AGE space equals space angle FGB space space left parenthesis opposite space angles right parenthesis rightwards double arrow angle FGB space equals space 65 degree Also comma space angle FGB space equals space angle HJI space space space space space space space space left parenthesis Corresponding space angle right parenthesis rightwards double arrow angle HJI space equals 65 degree Now comma space in space triangle HJI comma angle HJI space plus space angle JIH space plus space angle IHJ space equals space 180 degree rightwards double arrow 65 degree space plus space 40 degree space plus space angle IHJ space equals space 180 degree rightwards double arrow angle IHJ space equals space 180 degree space minus space 65 degree space minus space 40 degree space equals space 75 degree Now comma space straight x space equals space 180 degree space minus space angle IHJ space equals space 180 degree space minus space 75 degree equals space 105 degree Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 3

In figure, if lines l and m are parallel, then the value of x is

(a) 35°

(b) 55°

(c) 65°

(d) 75°

 

Solution 3

begin mathsize 12px style From space figure comma angle ACB space equals space 180 degree space minus space angle ACD equals space 180 degree space minus space 125 degree equals space 55 degree OR space angle BCA space equals 55 degree In space Right space triangle space ABC angle ABC space plus space angle BCA space plus space angle CAB space equals space 180 degree rightwards double arrow 90 degree space plus space 55 degree space plus space straight x space equals space 180 degree rightwards double arrow straight x space equals space 35 degree Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 4

Two complementary angles are such that two times the measure of one is equal to three times the measure of the other. The measure of the smaller angle is

(a) 45°

(b) 30°

(c) 36°

(d) none of these

Solution 4

Correct option (c)

Let one angle be θ

Then, its complementary = 90 - θ

According to question,

2θ = 3(90 - θ)

5θ = 270

θ = 54°

Then,  90 - θ° = 36°

Hence, the smaller angle is 36°.

Hence, correct option is (c).

Chapter 10 - Lines and Angles Excercise 10.57

Question 1

begin mathsize 12px style In space figure comma space if space straight y over straight x equals 5 space and space straight z over straight x equals 4 comma space then space the space value space of space straight x space is space left parenthesis straight a right parenthesis space 8 to the power of ring operator left parenthesis straight b right parenthesis space 18 to the power of ring operator left parenthesis straight c right parenthesis space 12 to the power of ring operator left parenthesis straight d right parenthesis space 15 to the power of ring operator end style

 

Solution 1

begin mathsize 12px style From space figure comma space we space can space see space that angle straight x degree space plus space angle straight y degree space plus space angle straight z degree space equals space space 180 degree space space space... left parenthesis 1 right parenthesis Now comma space straight y over straight x equals 5 rightwards double arrow space straight y space equals space 5 straight x And comma space straight z over straight x equals 4 space space space straight z space equals space 4 straight x Substituting space these space value space in space equation space left parenthesis 1 right parenthesis comma space we space have angle space straight x degree space plus space angle 5 straight x degree space plus space angle 4 straight x space equals space space 180 degree rightwards double arrow angle 10 straight x degree space equals space 180 degree rightwards double arrow angle straight x degree space equals space 18 degree space Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 2

begin mathsize 12px style In space figure comma space AB space parallel to space CD space parallel to space EF space and space GH space parallel to space KL. space The space measure space of space angle HKL space is left parenthesis straight a right parenthesis space 85 to the power of ring operator left parenthesis straight b right parenthesis space 135 to the power of ring operator left parenthesis straight c right parenthesis space 145 to the power of ring operator left parenthesis straight d right parenthesis space 215 to the power of ring operator end style

Solution 2

begin mathsize 12px style GH space parallel to space KL rightwards double arrow angle GHK space equals space angle HKL space space space space open parentheses interior space opposite space angles close parentheses Now comma space angle GHK space equals space angle GHD space plus space angle DHR space space space space space space space space space space space space space space space space space space space space space space space space space space equals space open parentheses 180 degree space minus space angle GHC close parentheses space plus space angle DHK space space space space left parenthesis angle GHC space and space angle GHD space are space supplementary right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space equals space 180 degree space minus space 60 degree space plus space 25 degree space rightwards double arrow angle GHK space equals space 145 degree rightwards double arrow angle HKL space equals space angle GHK space equals space 145 degree space Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 3

begin mathsize 12px style AB space and space CD space are space two space parallel space lines. space PQ space acuts space AB space and space CD space at space straight E space and space straight F space respectively. space EL space is space bisector space of angle FEB. space If space angle LEB space equals space 35 to the power of ring operator comma space then space angle CFQ space will space be left parenthesis straight a right parenthesis space 55 to the power of ring operator left parenthesis straight b right parenthesis space 70 to the power of ring operator left parenthesis straight c right parenthesis space 110 to the power of ring operator left parenthesis straight d right parenthesis space 130 to the power of ring operator end style

Solution 3

begin mathsize 12px style From space figure comma space angle LEB space equals space angle FEL space space space space space space space space left parenthesis EL space is space bisector space of space angle FEB right parenthesis Now comma space angle FEB space equals space 2 angle LEB space equals space 2 cross times 35 degree equals 70 degree Also comma space angle FEB space equals space angle CFE space space space space space left parenthesis Alternate space interior space angles right parenthesis rightwards double arrow angle CFE space equals space 70 degree Now comma space angle CFE space plus space angle CFQ space equals space 180 degree rightwards double arrow 70 degree space plus space angle CFQ space equals space 180 degree rightwards double arrow angle CFQ space equals space 180 degree space minus space 70 degree space equals space 110 degree Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 4

In figure, if line segment AB is parallel to the line segment CD, What is the value of y?

(a) 12

(b) 15

(c) 18

(d) 20


Solution 4

begin mathsize 12px style From space figure comma angle ABD space plus space angle EBD space equals space 180 degree space rightwards double arrow angle EBD space equals space 180 degree space minus space angle ABD space space space space space.... left parenthesis 1 right parenthesis Now comma space angle ABD space equals space straight y degree space plus space 2 straight y degree space plus space straight y degree space rightwards double arrow angle ABD equals space 4 straight y degree space space space.... left parenthesis 2 right parenthesis Substituting space left parenthesis 2 right parenthesis space in space left parenthesis 1 right parenthesis comma space we space have angle EBD space equals space 180 degree space minus space 4 straight y degree Now comma space angle EBD space equals space angle BDC space space space space space open parentheses Alternate space angles close parentheses rightwards double arrow 180 degree space minus space 4 straight y degree equals space 5 straight y degree rightwards double arrow 180 degree space equals space 9 straight y degree rightwards double arrow straight y space equals space 20 degree Hence comma space correct space option space is space left parenthesis straight d right parenthesis. end style

Question 5

begin mathsize 12px style In space figure comma space if space CP space parallel to space DQ comma space then space the space measure space of space straight x space is left parenthesis straight a right parenthesis space 130 to the power of ring operator left parenthesis straight b right parenthesis space 105 to the power of ring operator left parenthesis straight c right parenthesis space 175 to the power of ring operator left parenthesis straight d right parenthesis space 125 to the power of ring operator end style

Solution 5

begin mathsize 12px style From space Figure comma angle QBA space equals space angle CEA space space open parentheses Correspondence space angles close parentheses rightwards double arrow angle CEA equals 105 degree space space space space.... left parenthesis 1 right parenthesis In space triangle ACE comma angle CEA plus angle EAC plus angle ACE equals 180 degree rightwards double arrow 105 degree space plus space 25 degree space plus space angle ACE space equals space 180 degree space space space space space left square bracket from space left parenthesis 1 right parenthesis right square bracket rightwards double arrow 130 degree space plus space angle ACE space equals space 180 degree rightwards double arrow angle ACE space equals space 50 degree Now comma space straight x equals angle ACP equals 180 degree space minus space angle ACE equals 180 degree minus 50 degree equals 130 degree Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

CBSE Class 9 Maths Homework Help

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