Class 7 Exam > Class 7 Notes > Mathematics (Maths) Class 7 > RS Aggarwal Solutions: Lines & Angles
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Page 1
Q u e s t i o n : 1
Find the complement of each of the following angles:
i
35°
ii
47°
iii
60°
iv
73°
S o l u t i o n :
i
The given angle measures 35°.
Let the measure of its complement be x.
x + 35° = 90°
or x = 90 -35
° = 55°
Hence, the complement of the given angle will be 55°.
ii
The given angle measures 47°.
Let the measure of its complement be x.
x + 47° = 90°
or x = 90 -47
° = 43°
Hence, the complement of the given angle will be 43°.
iii
The given angle measures 60°.
Let the measure of its complement be x°.
x + 60° = 90°
or x = 90 -60
° = 30°
Hence, the complement of the given angle will be 30°.
iv
The given angle measures 73°.
Let the measure of its complement be x.
x + 73
o
= 90°
or x = 90 -73
° = 17°
Hence, the complement of the given angle will be 17°.
Q u e s t i o n : 2
Find the supplement of each of the following angles:
i
80°
ii
54°
iii
105°
iv
123°
S o l u t i o n :
i
The given angle measures 80°.
Let the measure of its supplement be x.
x + 80° = 180°
or x = 180 -80
° = 100°
Hence, the complement of the given angle will be 100°.
ii
The given angle measures 54°.
Let the measure of its supplement be x.
x + 54° = 180°
or x = 180 -54
° = 126°
Hence, the complement of the given angle will be 126°.
iii
The given angle measures 105°.
Let the measure of its supplement be x.
x + 105° = 180°
or, x = 180 -105
° = 75°
Hence, the complement of the given angle will be 75°.
iv
The given angle measures 123°.
Let the measure of its supplement be x.
Page 2
Q u e s t i o n : 1
Find the complement of each of the following angles:
i
35°
ii
47°
iii
60°
iv
73°
S o l u t i o n :
i
The given angle measures 35°.
Let the measure of its complement be x.
x + 35° = 90°
or x = 90 -35
° = 55°
Hence, the complement of the given angle will be 55°.
ii
The given angle measures 47°.
Let the measure of its complement be x.
x + 47° = 90°
or x = 90 -47
° = 43°
Hence, the complement of the given angle will be 43°.
iii
The given angle measures 60°.
Let the measure of its complement be x°.
x + 60° = 90°
or x = 90 -60
° = 30°
Hence, the complement of the given angle will be 30°.
iv
The given angle measures 73°.
Let the measure of its complement be x.
x + 73
o
= 90°
or x = 90 -73
° = 17°
Hence, the complement of the given angle will be 17°.
Q u e s t i o n : 2
Find the supplement of each of the following angles:
i
80°
ii
54°
iii
105°
iv
123°
S o l u t i o n :
i
The given angle measures 80°.
Let the measure of its supplement be x.
x + 80° = 180°
or x = 180 -80
° = 100°
Hence, the complement of the given angle will be 100°.
ii
The given angle measures 54°.
Let the measure of its supplement be x.
x + 54° = 180°
or x = 180 -54
° = 126°
Hence, the complement of the given angle will be 126°.
iii
The given angle measures 105°.
Let the measure of its supplement be x.
x + 105° = 180°
or, x = 180 -105
° = 75°
Hence, the complement of the given angle will be 75°.
iv
The given angle measures 123°.
Let the measure of its supplement be x.
x + 123° = 180°
or x = 180 -123
° = 57°
Hence, the complement of the given angle will be 57°.
Q u e s t i o n : 3
Among two supplementary angles, the measure of the larger angle is 36° more than the measure of the smaller. Find their measures.
S o l u t i o n :
Let the two supplementary angles be x°
and 180 -x
°.
Since it is given that the measure of the larger angle is 36° more than the smaller angle, let the larger angle be x°.
? 180 -x
°
+ 36° = x°
or 216 = 2x
or 108 = x
Larger angle = 108°
Smaller angle = 108 -36
°
= 72°
Q u e s t i o n : 4
Find the angle which is equal to its supplement.
S o l u t i o n :
Let the measure of the required angle be x.
Since it is its own supplement:
x + x = 180°or 2x = 180°or x = 90°
Therefore, the required angle is 90°.
Q u e s t i o n : 5
Can two angles be supplementary if both of them are:
i
acute?
ii
obtuse?
iii
right?
S o l u t i o n :
i
No. If both the angles are acute, i.e. less than 90°, they cannot be supplementary as their sum will always be less than 180°.
ii
No. If both the angles are obtuse, i.e. more than 90°, they cannot be supplementary as their sum will always be more than 180°.
iii
Yes. If both the angles are right, i.e. they both measure 90°, then they form a supplementary pair.
90°
+ 90° = 180°
Q u e s t i o n : 6
In the given figure, AOB is a straight line and the ray OC stands on it.
If ?AOC = 64° and ?BOC = x°, find the value of x.
S o l u t i o n :
By linear pair property:
?AOC + ?COB = 180°64° + ?COB = 180° ?COB = x° = 180° - 64° = 116°
? x = 116
Q u e s t i o n : 7
In the given figure, AOB is a straight line and the ray OC stands on it.
If ?AOC = (2x - 10)° and ?BOC = (3x + 20)°, find the value of x.
Also, find ?AOC and ?BOC
S o l u t i o n :
° ) 0 2 + x 3 ( = C O B ? , ° ) 0 1 – x 2 ( = C O A ? t i n o s d n a t s C O y a r d n a e n i l t h g i a r t s a s i B O A
° 2 2 1 = ° 0 2 + ° 2 0 1 – ° 0 2 + ° ( 3 x 3 = ° ) 0 2 + x 3 ( = C O B ? ° 8 5 = ° 0 1 – ° 8 ) = 0 1 – ° ( 3 x 2 = ° ) 0 1 – x 2 ( = C O A ? ° ( 3 = x ? ° 0 7 1 = x 5 ? ° 0 8 1 = ° 0 1 + x 5 ? ° 0 8 1 = ° 0 2 + x 3 + ° 0 1 – x 2 ? ) r i a p r a e n i L ( ° 0 8 1 = C O B ? + C O A ?
Page 3
Q u e s t i o n : 1
Find the complement of each of the following angles:
i
35°
ii
47°
iii
60°
iv
73°
S o l u t i o n :
i
The given angle measures 35°.
Let the measure of its complement be x.
x + 35° = 90°
or x = 90 -35
° = 55°
Hence, the complement of the given angle will be 55°.
ii
The given angle measures 47°.
Let the measure of its complement be x.
x + 47° = 90°
or x = 90 -47
° = 43°
Hence, the complement of the given angle will be 43°.
iii
The given angle measures 60°.
Let the measure of its complement be x°.
x + 60° = 90°
or x = 90 -60
° = 30°
Hence, the complement of the given angle will be 30°.
iv
The given angle measures 73°.
Let the measure of its complement be x.
x + 73
o
= 90°
or x = 90 -73
° = 17°
Hence, the complement of the given angle will be 17°.
Q u e s t i o n : 2
Find the supplement of each of the following angles:
i
80°
ii
54°
iii
105°
iv
123°
S o l u t i o n :
i
The given angle measures 80°.
Let the measure of its supplement be x.
x + 80° = 180°
or x = 180 -80
° = 100°
Hence, the complement of the given angle will be 100°.
ii
The given angle measures 54°.
Let the measure of its supplement be x.
x + 54° = 180°
or x = 180 -54
° = 126°
Hence, the complement of the given angle will be 126°.
iii
The given angle measures 105°.
Let the measure of its supplement be x.
x + 105° = 180°
or, x = 180 -105
° = 75°
Hence, the complement of the given angle will be 75°.
iv
The given angle measures 123°.
Let the measure of its supplement be x.
x + 123° = 180°
or x = 180 -123
° = 57°
Hence, the complement of the given angle will be 57°.
Q u e s t i o n : 3
Among two supplementary angles, the measure of the larger angle is 36° more than the measure of the smaller. Find their measures.
S o l u t i o n :
Let the two supplementary angles be x°
and 180 -x
°.
Since it is given that the measure of the larger angle is 36° more than the smaller angle, let the larger angle be x°.
? 180 -x
°
+ 36° = x°
or 216 = 2x
or 108 = x
Larger angle = 108°
Smaller angle = 108 -36
°
= 72°
Q u e s t i o n : 4
Find the angle which is equal to its supplement.
S o l u t i o n :
Let the measure of the required angle be x.
Since it is its own supplement:
x + x = 180°or 2x = 180°or x = 90°
Therefore, the required angle is 90°.
Q u e s t i o n : 5
Can two angles be supplementary if both of them are:
i
acute?
ii
obtuse?
iii
right?
S o l u t i o n :
i
No. If both the angles are acute, i.e. less than 90°, they cannot be supplementary as their sum will always be less than 180°.
ii
No. If both the angles are obtuse, i.e. more than 90°, they cannot be supplementary as their sum will always be more than 180°.
iii
Yes. If both the angles are right, i.e. they both measure 90°, then they form a supplementary pair.
90°
+ 90° = 180°
Q u e s t i o n : 6
In the given figure, AOB is a straight line and the ray OC stands on it.
If ?AOC = 64° and ?BOC = x°, find the value of x.
S o l u t i o n :
By linear pair property:
?AOC + ?COB = 180°64° + ?COB = 180° ?COB = x° = 180° - 64° = 116°
? x = 116
Q u e s t i o n : 7
In the given figure, AOB is a straight line and the ray OC stands on it.
If ?AOC = (2x - 10)° and ?BOC = (3x + 20)°, find the value of x.
Also, find ?AOC and ?BOC
S o l u t i o n :
° ) 0 2 + x 3 ( = C O B ? , ° ) 0 1 – x 2 ( = C O A ? t i n o s d n a t s C O y a r d n a e n i l t h g i a r t s a s i B O A
° 2 2 1 = ° 0 2 + ° 2 0 1 – ° 0 2 + ° ( 3 x 3 = ° ) 0 2 + x 3 ( = C O B ? ° 8 5 = ° 0 1 – ° 8 ) = 0 1 – ° ( 3 x 2 = ° ) 0 1 – x 2 ( = C O A ? ° ( 3 = x ? ° 0 7 1 = x 5 ? ° 0 8 1 = ° 0 1 + x 5 ? ° 0 8 1 = ° 0 2 + x 3 + ° 0 1 – x 2 ? ) r i a p r a e n i L ( ° 0 8 1 = C O B ? + C O A ?
Q u e s t i o n : 8
In the given figure, AOB is a straight line and the rays OC and OD stands on it.
If ?AOC = 65°, ?BOD = 70° and ?COD = x° find the value of x.
S o l u t i o n :
Since AOB is a straight line, we have:
?AOC + ?BOD + ?COD = 180°or 65°+ 70° + x° = 180° (given)or 135° + x° = 180°or x° = 45°Thus, the value of x is 45
Q u e s t i o n : 9
In the given figure, two straight line AB and CD intersect at a point O.
If ?AOC = 42°, find the measure of each of the angles:
i
?AOD
ii
?BOD
iii
?COB
S o l u t i o n :
? OD
Q u e s t i o n : 1 0
In the given figure, two straight line PQ and RS intersect at a O.
If ?POS = 114°, find the measure of each of the angles:
i
?POR
ii
?ROQ
iii
?QOS
S o l u t i o n :
Q u e s t i o n : 1 1
In the given figure, rays OA, OB, OC and OD are such that
Page 4
Q u e s t i o n : 1
Find the complement of each of the following angles:
i
35°
ii
47°
iii
60°
iv
73°
S o l u t i o n :
i
The given angle measures 35°.
Let the measure of its complement be x.
x + 35° = 90°
or x = 90 -35
° = 55°
Hence, the complement of the given angle will be 55°.
ii
The given angle measures 47°.
Let the measure of its complement be x.
x + 47° = 90°
or x = 90 -47
° = 43°
Hence, the complement of the given angle will be 43°.
iii
The given angle measures 60°.
Let the measure of its complement be x°.
x + 60° = 90°
or x = 90 -60
° = 30°
Hence, the complement of the given angle will be 30°.
iv
The given angle measures 73°.
Let the measure of its complement be x.
x + 73
o
= 90°
or x = 90 -73
° = 17°
Hence, the complement of the given angle will be 17°.
Q u e s t i o n : 2
Find the supplement of each of the following angles:
i
80°
ii
54°
iii
105°
iv
123°
S o l u t i o n :
i
The given angle measures 80°.
Let the measure of its supplement be x.
x + 80° = 180°
or x = 180 -80
° = 100°
Hence, the complement of the given angle will be 100°.
ii
The given angle measures 54°.
Let the measure of its supplement be x.
x + 54° = 180°
or x = 180 -54
° = 126°
Hence, the complement of the given angle will be 126°.
iii
The given angle measures 105°.
Let the measure of its supplement be x.
x + 105° = 180°
or, x = 180 -105
° = 75°
Hence, the complement of the given angle will be 75°.
iv
The given angle measures 123°.
Let the measure of its supplement be x.
x + 123° = 180°
or x = 180 -123
° = 57°
Hence, the complement of the given angle will be 57°.
Q u e s t i o n : 3
Among two supplementary angles, the measure of the larger angle is 36° more than the measure of the smaller. Find their measures.
S o l u t i o n :
Let the two supplementary angles be x°
and 180 -x
°.
Since it is given that the measure of the larger angle is 36° more than the smaller angle, let the larger angle be x°.
? 180 -x
°
+ 36° = x°
or 216 = 2x
or 108 = x
Larger angle = 108°
Smaller angle = 108 -36
°
= 72°
Q u e s t i o n : 4
Find the angle which is equal to its supplement.
S o l u t i o n :
Let the measure of the required angle be x.
Since it is its own supplement:
x + x = 180°or 2x = 180°or x = 90°
Therefore, the required angle is 90°.
Q u e s t i o n : 5
Can two angles be supplementary if both of them are:
i
acute?
ii
obtuse?
iii
right?
S o l u t i o n :
i
No. If both the angles are acute, i.e. less than 90°, they cannot be supplementary as their sum will always be less than 180°.
ii
No. If both the angles are obtuse, i.e. more than 90°, they cannot be supplementary as their sum will always be more than 180°.
iii
Yes. If both the angles are right, i.e. they both measure 90°, then they form a supplementary pair.
90°
+ 90° = 180°
Q u e s t i o n : 6
In the given figure, AOB is a straight line and the ray OC stands on it.
If ?AOC = 64° and ?BOC = x°, find the value of x.
S o l u t i o n :
By linear pair property:
?AOC + ?COB = 180°64° + ?COB = 180° ?COB = x° = 180° - 64° = 116°
? x = 116
Q u e s t i o n : 7
In the given figure, AOB is a straight line and the ray OC stands on it.
If ?AOC = (2x - 10)° and ?BOC = (3x + 20)°, find the value of x.
Also, find ?AOC and ?BOC
S o l u t i o n :
° ) 0 2 + x 3 ( = C O B ? , ° ) 0 1 – x 2 ( = C O A ? t i n o s d n a t s C O y a r d n a e n i l t h g i a r t s a s i B O A
° 2 2 1 = ° 0 2 + ° 2 0 1 – ° 0 2 + ° ( 3 x 3 = ° ) 0 2 + x 3 ( = C O B ? ° 8 5 = ° 0 1 – ° 8 ) = 0 1 – ° ( 3 x 2 = ° ) 0 1 – x 2 ( = C O A ? ° ( 3 = x ? ° 0 7 1 = x 5 ? ° 0 8 1 = ° 0 1 + x 5 ? ° 0 8 1 = ° 0 2 + x 3 + ° 0 1 – x 2 ? ) r i a p r a e n i L ( ° 0 8 1 = C O B ? + C O A ?
Q u e s t i o n : 8
In the given figure, AOB is a straight line and the rays OC and OD stands on it.
If ?AOC = 65°, ?BOD = 70° and ?COD = x° find the value of x.
S o l u t i o n :
Since AOB is a straight line, we have:
?AOC + ?BOD + ?COD = 180°or 65°+ 70° + x° = 180° (given)or 135° + x° = 180°or x° = 45°Thus, the value of x is 45
Q u e s t i o n : 9
In the given figure, two straight line AB and CD intersect at a point O.
If ?AOC = 42°, find the measure of each of the angles:
i
?AOD
ii
?BOD
iii
?COB
S o l u t i o n :
? OD
Q u e s t i o n : 1 0
In the given figure, two straight line PQ and RS intersect at a O.
If ?POS = 114°, find the measure of each of the angles:
i
?POR
ii
?ROQ
iii
?QOS
S o l u t i o n :
Q u e s t i o n : 1 1
In the given figure, rays OA, OB, OC and OD are such that
?AOB = 56°, ?BOC = 100°, ?COD = x° and ?DOA = 74°.
Find the value of x.
S o l u t i o n :
Sum of all the angles around a point is 360°.
? ?AOB + ?BOC + ? COD + ?DOA = 360°or 56° + 100° + x° + 74° = 360° (given)or 230° + x° = 360°or x° = 130°or x = 130
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