Lines & Angles (Exercise 14.2) RD Sharma Solutions | Mathematics (Maths) Class 7 PDF Download

 Page 1


 
 
 
 
 
 
 
        
 
1. In Fig. 58, line n is a transversal to line l and m. Identify the following: 
(i) Alternate and corresponding angles in Fig. 58 (i) 
(ii) Angles alternate to ?d and ?g and angles corresponding to ?f and ?h in Fig. 58 (ii) 
(iii) Angle alternate to ?PQR, angle corresponding to ?RQF and angle alternate to 
?PQE in Fig. 58 (iii) 
(iv) Pairs of interior and exterior angles on the same side of the transversal in Fig. 58 
(ii) 
  
 
Solution: 
(i) A pair of angles in which one arm of both the angles is on the same side of the 
transversal and their other arms are directed in the same sense is called a pair of 
corresponding angles. 
In Figure (i) Corresponding angles are 
?EGB and ?GHD 
?HGB and ?FHD 
?EGA and ?GHC 
?AGH and ?CHF 
A pair of angles in which one arm of each of the angle is on opposite sides of the 
transversal and whose other arms include the one segment is called a pair of alternate 
angles. 
The alternate angles are: 
?EGB and ?CHF 
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1. In Fig. 58, line n is a transversal to line l and m. Identify the following: 
(i) Alternate and corresponding angles in Fig. 58 (i) 
(ii) Angles alternate to ?d and ?g and angles corresponding to ?f and ?h in Fig. 58 (ii) 
(iii) Angle alternate to ?PQR, angle corresponding to ?RQF and angle alternate to 
?PQE in Fig. 58 (iii) 
(iv) Pairs of interior and exterior angles on the same side of the transversal in Fig. 58 
(ii) 
  
 
Solution: 
(i) A pair of angles in which one arm of both the angles is on the same side of the 
transversal and their other arms are directed in the same sense is called a pair of 
corresponding angles. 
In Figure (i) Corresponding angles are 
?EGB and ?GHD 
?HGB and ?FHD 
?EGA and ?GHC 
?AGH and ?CHF 
A pair of angles in which one arm of each of the angle is on opposite sides of the 
transversal and whose other arms include the one segment is called a pair of alternate 
angles. 
The alternate angles are: 
?EGB and ?CHF 
 
 
 
 
 
 
 
?HGB and ?CHG 
?EGA and ?FHD 
?AGH and ?GHD 
 
(ii) In Figure (ii) 
The alternate angle to ?d is ?e. 
The alternate angle to ?g is ?b. 
The corresponding angle to ?f is ?c. 
The corresponding angle to ?h is ?a. 
 
(iii) In Figure (iii) 
Angle alternate to ?PQR is ?QRA. 
Angle corresponding to ?RQF is ?ARB. 
Angle alternate to ?POE is ?ARB. 
 
(iv) In Figure (ii) 
Pair of interior angles are 
?a is ?e. 
?d is ?f. 
Pair of exterior angles are 
?b is ?h. 
?c is ?g. 
 
2. In Fig. 59, AB and CD are parallel lines intersected by a transversal PQ at L and M 
respectively, If ?CMQ = 60
o
, find all other angles in the figure. 
   
Solution: 
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1. In Fig. 58, line n is a transversal to line l and m. Identify the following: 
(i) Alternate and corresponding angles in Fig. 58 (i) 
(ii) Angles alternate to ?d and ?g and angles corresponding to ?f and ?h in Fig. 58 (ii) 
(iii) Angle alternate to ?PQR, angle corresponding to ?RQF and angle alternate to 
?PQE in Fig. 58 (iii) 
(iv) Pairs of interior and exterior angles on the same side of the transversal in Fig. 58 
(ii) 
  
 
Solution: 
(i) A pair of angles in which one arm of both the angles is on the same side of the 
transversal and their other arms are directed in the same sense is called a pair of 
corresponding angles. 
In Figure (i) Corresponding angles are 
?EGB and ?GHD 
?HGB and ?FHD 
?EGA and ?GHC 
?AGH and ?CHF 
A pair of angles in which one arm of each of the angle is on opposite sides of the 
transversal and whose other arms include the one segment is called a pair of alternate 
angles. 
The alternate angles are: 
?EGB and ?CHF 
 
 
 
 
 
 
 
?HGB and ?CHG 
?EGA and ?FHD 
?AGH and ?GHD 
 
(ii) In Figure (ii) 
The alternate angle to ?d is ?e. 
The alternate angle to ?g is ?b. 
The corresponding angle to ?f is ?c. 
The corresponding angle to ?h is ?a. 
 
(iii) In Figure (iii) 
Angle alternate to ?PQR is ?QRA. 
Angle corresponding to ?RQF is ?ARB. 
Angle alternate to ?POE is ?ARB. 
 
(iv) In Figure (ii) 
Pair of interior angles are 
?a is ?e. 
?d is ?f. 
Pair of exterior angles are 
?b is ?h. 
?c is ?g. 
 
2. In Fig. 59, AB and CD are parallel lines intersected by a transversal PQ at L and M 
respectively, If ?CMQ = 60
o
, find all other angles in the figure. 
   
Solution: 
 
 
 
 
 
 
 
A pair of angles in which one arm of both the angles is on the same side of the 
transversal and their other arms are directed in the same sense is called a pair of 
corresponding angles. 
Therefore corresponding angles are 
?ALM = ?CMQ = 60
o
 [given] 
Vertically opposite angles are 
?LMD = ?CMQ = 60
o 
[given] 
Vertically opposite angles are  
?ALM = ?PLB = 60
o 
Here, ?CMQ + ?QMD = 180
o
 are the linear pair 
On rearranging we get 
?QMD = 180
o
 – 60
o 
= 120
o 
Corresponding angles are 
?QMD = ?MLB = 120
o 
Vertically opposite angles 
?QMD = ?CML = 120
o 
Vertically opposite angles 
?MLB = ?ALP = 120
o
 
 
3. In Fig. 60, AB and CD are parallel lines intersected by a transversal by a transversal 
PQ at L and M respectively. If ?LMD = 35
o
 find ?ALM and ?PLA. 
 
Solution: 
Given that, ?LMD = 35
o 
From the figure we can write 
?LMD and ?LMC is a linear pair 
?LMD + ?LMC = 180
o 
[sum of angles in linear pair = 180
o
] 
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1. In Fig. 58, line n is a transversal to line l and m. Identify the following: 
(i) Alternate and corresponding angles in Fig. 58 (i) 
(ii) Angles alternate to ?d and ?g and angles corresponding to ?f and ?h in Fig. 58 (ii) 
(iii) Angle alternate to ?PQR, angle corresponding to ?RQF and angle alternate to 
?PQE in Fig. 58 (iii) 
(iv) Pairs of interior and exterior angles on the same side of the transversal in Fig. 58 
(ii) 
  
 
Solution: 
(i) A pair of angles in which one arm of both the angles is on the same side of the 
transversal and their other arms are directed in the same sense is called a pair of 
corresponding angles. 
In Figure (i) Corresponding angles are 
?EGB and ?GHD 
?HGB and ?FHD 
?EGA and ?GHC 
?AGH and ?CHF 
A pair of angles in which one arm of each of the angle is on opposite sides of the 
transversal and whose other arms include the one segment is called a pair of alternate 
angles. 
The alternate angles are: 
?EGB and ?CHF 
 
 
 
 
 
 
 
?HGB and ?CHG 
?EGA and ?FHD 
?AGH and ?GHD 
 
(ii) In Figure (ii) 
The alternate angle to ?d is ?e. 
The alternate angle to ?g is ?b. 
The corresponding angle to ?f is ?c. 
The corresponding angle to ?h is ?a. 
 
(iii) In Figure (iii) 
Angle alternate to ?PQR is ?QRA. 
Angle corresponding to ?RQF is ?ARB. 
Angle alternate to ?POE is ?ARB. 
 
(iv) In Figure (ii) 
Pair of interior angles are 
?a is ?e. 
?d is ?f. 
Pair of exterior angles are 
?b is ?h. 
?c is ?g. 
 
2. In Fig. 59, AB and CD are parallel lines intersected by a transversal PQ at L and M 
respectively, If ?CMQ = 60
o
, find all other angles in the figure. 
   
Solution: 
 
 
 
 
 
 
 
A pair of angles in which one arm of both the angles is on the same side of the 
transversal and their other arms are directed in the same sense is called a pair of 
corresponding angles. 
Therefore corresponding angles are 
?ALM = ?CMQ = 60
o
 [given] 
Vertically opposite angles are 
?LMD = ?CMQ = 60
o 
[given] 
Vertically opposite angles are  
?ALM = ?PLB = 60
o 
Here, ?CMQ + ?QMD = 180
o
 are the linear pair 
On rearranging we get 
?QMD = 180
o
 – 60
o 
= 120
o 
Corresponding angles are 
?QMD = ?MLB = 120
o 
Vertically opposite angles 
?QMD = ?CML = 120
o 
Vertically opposite angles 
?MLB = ?ALP = 120
o
 
 
3. In Fig. 60, AB and CD are parallel lines intersected by a transversal by a transversal 
PQ at L and M respectively. If ?LMD = 35
o
 find ?ALM and ?PLA. 
 
Solution: 
Given that, ?LMD = 35
o 
From the figure we can write 
?LMD and ?LMC is a linear pair 
?LMD + ?LMC = 180
o 
[sum of angles in linear pair = 180
o
] 
 
 
 
 
 
 
 
On rearranging, we get 
?LMC = 180
o
 – 35
o 
= 145
o 
So, ?LMC = ?PLA = 145
o 
And, ?LMC = ?MLB = 145
o 
?MLB and ?ALM is a linear pair 
?MLB + ?ALM = 180
o
 [sum of angles in linear pair = 180
o
] 
?ALM = 180
o
 – 145
o 
?ALM = 35
0 
Therefore, ?ALM = 35
o
, ?PLA = 145
o
. 
 
4. The line n is transversal to line l and m in Fig. 61. Identify the angle alternate to ?13, 
angle corresponding to ?15, and angle alternate to ?15. 
    
 
Solution: 
Given that, l ? m 
From the figure the angle alternate to ?13 is ?7 
From the figure the angle corresponding to ?15 is ?7 [A pair of angles in which one arm 
of both the angles is on the same side of the transversal and their other arms are 
directed in the same sense is called a pair of corresponding angles.] 
Again from the figure angle alternate to ?15 is ?5 
 
5. In Fig. 62, line l ? m and n is transversal. If ?1 = 40°, find all the angles and check 
Page 5


 
 
 
 
 
 
 
        
 
1. In Fig. 58, line n is a transversal to line l and m. Identify the following: 
(i) Alternate and corresponding angles in Fig. 58 (i) 
(ii) Angles alternate to ?d and ?g and angles corresponding to ?f and ?h in Fig. 58 (ii) 
(iii) Angle alternate to ?PQR, angle corresponding to ?RQF and angle alternate to 
?PQE in Fig. 58 (iii) 
(iv) Pairs of interior and exterior angles on the same side of the transversal in Fig. 58 
(ii) 
  
 
Solution: 
(i) A pair of angles in which one arm of both the angles is on the same side of the 
transversal and their other arms are directed in the same sense is called a pair of 
corresponding angles. 
In Figure (i) Corresponding angles are 
?EGB and ?GHD 
?HGB and ?FHD 
?EGA and ?GHC 
?AGH and ?CHF 
A pair of angles in which one arm of each of the angle is on opposite sides of the 
transversal and whose other arms include the one segment is called a pair of alternate 
angles. 
The alternate angles are: 
?EGB and ?CHF 
 
 
 
 
 
 
 
?HGB and ?CHG 
?EGA and ?FHD 
?AGH and ?GHD 
 
(ii) In Figure (ii) 
The alternate angle to ?d is ?e. 
The alternate angle to ?g is ?b. 
The corresponding angle to ?f is ?c. 
The corresponding angle to ?h is ?a. 
 
(iii) In Figure (iii) 
Angle alternate to ?PQR is ?QRA. 
Angle corresponding to ?RQF is ?ARB. 
Angle alternate to ?POE is ?ARB. 
 
(iv) In Figure (ii) 
Pair of interior angles are 
?a is ?e. 
?d is ?f. 
Pair of exterior angles are 
?b is ?h. 
?c is ?g. 
 
2. In Fig. 59, AB and CD are parallel lines intersected by a transversal PQ at L and M 
respectively, If ?CMQ = 60
o
, find all other angles in the figure. 
   
Solution: 
 
 
 
 
 
 
 
A pair of angles in which one arm of both the angles is on the same side of the 
transversal and their other arms are directed in the same sense is called a pair of 
corresponding angles. 
Therefore corresponding angles are 
?ALM = ?CMQ = 60
o
 [given] 
Vertically opposite angles are 
?LMD = ?CMQ = 60
o 
[given] 
Vertically opposite angles are  
?ALM = ?PLB = 60
o 
Here, ?CMQ + ?QMD = 180
o
 are the linear pair 
On rearranging we get 
?QMD = 180
o
 – 60
o 
= 120
o 
Corresponding angles are 
?QMD = ?MLB = 120
o 
Vertically opposite angles 
?QMD = ?CML = 120
o 
Vertically opposite angles 
?MLB = ?ALP = 120
o
 
 
3. In Fig. 60, AB and CD are parallel lines intersected by a transversal by a transversal 
PQ at L and M respectively. If ?LMD = 35
o
 find ?ALM and ?PLA. 
 
Solution: 
Given that, ?LMD = 35
o 
From the figure we can write 
?LMD and ?LMC is a linear pair 
?LMD + ?LMC = 180
o 
[sum of angles in linear pair = 180
o
] 
 
 
 
 
 
 
 
On rearranging, we get 
?LMC = 180
o
 – 35
o 
= 145
o 
So, ?LMC = ?PLA = 145
o 
And, ?LMC = ?MLB = 145
o 
?MLB and ?ALM is a linear pair 
?MLB + ?ALM = 180
o
 [sum of angles in linear pair = 180
o
] 
?ALM = 180
o
 – 145
o 
?ALM = 35
0 
Therefore, ?ALM = 35
o
, ?PLA = 145
o
. 
 
4. The line n is transversal to line l and m in Fig. 61. Identify the angle alternate to ?13, 
angle corresponding to ?15, and angle alternate to ?15. 
    
 
Solution: 
Given that, l ? m 
From the figure the angle alternate to ?13 is ?7 
From the figure the angle corresponding to ?15 is ?7 [A pair of angles in which one arm 
of both the angles is on the same side of the transversal and their other arms are 
directed in the same sense is called a pair of corresponding angles.] 
Again from the figure angle alternate to ?15 is ?5 
 
5. In Fig. 62, line l ? m and n is transversal. If ?1 = 40°, find all the angles and check 
 
 
 
 
 
 
 
that all corresponding angles and alternate angles are equal. 
    
 
Solution: 
Given that, ?1 = 40
o 
?1 and ?2 is a linear pair [from the figure] 
?1 + ?2 = 180
o 
?2 = 180
o
 – 40
o 
?2 = 140
o
 
Again from the figure we can say that 
?2 and ?6 is a corresponding angle pair 
So, ?6 = 140
o 
?6 and ?5 is a linear pair [from the figure] 
?6 + ?5 = 180
o
 
?5 = 180
o
 – 140
o 
?5 = 40
o 
From the figure we can write as 
?3 and ?5 are alternate interior angles 
So, ?5 = ?3 = 40
o 
?3 and ?4 is a linear pair 
?3 + ?4 = 180
o 
?4 = 180
o
 – 40
o 
?4 = 140
o
 
Now, ?4 and ?6 are a pair of interior angles 
So, ?4 = ?6 = 140
o 
?3 and ?7 are a pair of corresponding angles 
So, ?3 = ?7 = 40
o 
Therefore, ?7 = 40
o 
?4 and ?8 are a pair of corresponding angles 
So, ?4 = ?8 = 140
o