Class – 7 CH-5 LINES AND ANGLES
MATHS NCERT SOLUTIONS
Exercise 5.1
Question 1:
Find the complement of each of the following angles:
Question 2:
Find the supplement of each of the following angles:
3
Identify which of the following pairs of angles are complementary and which are supplementary:
(i) 65 ,115 (ii) 63,27 (iii) 112 ,68
(iv) 130 ,50 (v) 45,45 (vi) 80,10
SOLUTION 3:
If sum of two angles is 180, then they are called supplementary angles.
If sum of two angles is 90, then they are called complementary angles.
(i) 65115180 These are supplementary angles.
(ii) 63 27 90 These are complementary angles.
(iii) 112 68 180 These are supplementary angles.
(iv) 130 50 180 These are supplementary angles.
(v) 45 45 90 These are complementary angles. (vi) 80 10 90 These are complementary angles.
Question 4: Find the angle which is equal to its complement.
SOLUTION 4:
Let one of the two equal complementary angles be x.
x x 90
2x 90
x 45
Thus, 45 is equal to its complement.
Question 5: Find the angle which is equal to its supplement. SOLUTION 5:
Let x be two equal angles of its supplement.
Therefore, x x 180 [Supplementary angles]
2x 180
180
x 90
2
Thus, 90 is equal to its supplement.
6
In the given figure, 1 and 2 are supplementary angles. If 1 is decreased, what changes should take place in 2 so that both the angles still remain supplementary?
SOLUTION 6:
If 1 is decreased then, 2 will increase with the same measure, so that both the angles still remain supplementary.
Question 7:
Can two angles be supplementary if both of them are:
(i) acute (ii) obtuse (iii) right?
SOLUTION 7:
(i) No, because sum of two acute angles is less than 180 .
(ii) No, because sum of two obtuse angles is more than 180 .
(iii) Yes, because sum of two right angles is 180 .
Question 8:
An angle is greater than 45 . Is its complementary angle greater than 45 or equal to 45 or less than 45 ?
SOLUTION 8:
Let the complementary angles be x and y, i.e., x y 90
It is given that x 45
Adding y both sides, x y 45 y
90 45 y
90 45 y
y 45
Thus, its complementary angle is less than 45 .
9
In the adjoining figure:
(i) Is 1 adjacent to 2?
(ii) Is AOC adjacent to AOE?
(iii) Do COE and EOD form a linear pair?
(iv) Are BOD and DOA supplementary?
(v) Is 1 vertically opposite to 4? (vi) What is the vertically opposite angle of 5?
SOLUTION 9:
(i) Yes, in AOE, OC is common arm.
(ii) No, they have no non-common arms on opposite side of common arm.
(iii) Yes, they form linear pair.
(iv) Yes, they are supplementary.
(v) Yes, they are vertically opposite angles.
(vi) Vertically opposite angles of 5 is COB.
Question 10:
Indicate which pairs of angles are:
(i) Vertically opposite angles? (ii) Linear pairs?
SOLUTION 10:
(i) Vertically opposite angles, 1 and 4; 5 and 2 + 3. (ii) Linear pairs 1 and 5; 5 and 4.
Question 11:
In the following figure, is 1 adjacent to 2? Give reasons.
SOLUTION 11:
1 and 2 are not adjacent angles because their vertex is not common.
Question 12:
Find the values of the angles x y, and z in each of the following:
(i) (ii)
SOLUTION 12:
(i) x 55 [Vertically opposite angles]
Now 55 y 180 [Linear pair]
y 180 55 125
Also y z 125
Thus, x 55 , y 125 and z 125 .
[Vertically opposite angles]
(ii) 40 x 25 180 [Angles on straight line]
65 x 180
x 180 65 = 115
Now 40 y 180 [Linear pair]
y 180 40 140 ……….(i)
Also y z 180 [Linear pair]
140 z 180 [From equation (i)]
z 180140 40
Thus, x 115 , y 140 and z 40 .
Question 13:
Fill in the blanks:
(i) If two angles are complementary, then the sum of their measures is
_______________.
(ii) If two angles are supplementary, then the sum of their measures is
_______________.
(iii) Two angles forming a linear pair are _______________.
(iv) If two adjacent angles are supplementary, they form a _______________.
(v) If two lines intersect a point, then the vertically opposite angles are always _______________.
(vi) If two lines intersect at a point and if one pair of vertically opposite angles are acute angles, then the other pair of vertically opposite angles are
_______________.
SOLUTION 13:
(i) 90 (ii) 180 (iii) supplementary
(iv) linear pair (v) equal (vi) obtuse angles
Question 14:
In the adjoining figure, name the following pairs of angles:
(i) Obtuse vertically opposite angles.
(ii) Adjacent complementary angles.
(iii) Equal supplementary angles.
(iv) Unequal supplementary angles. (v) Adjacent angles that do not form a linear pair.
SOLUTION 14:
(i) Obtuse vertically opposite angles means greater than 90 and equal AOD = BOC.
(ii) Adjacent complementary angles means angles have common vertex, common arm, non-common arms are on either side of common arm and sum of angles is 90 .
(iii) Equal supplementary angles means sum of angles is 180 and supplement angles are equal.
(iv) Unequal supplementary angles means sum of angles is 180 and supplement angles are unequal.
i. e., AOE, EOC; AOD, DOC and AOB, BOC
(v) Adjacent angles that do not form a linear pair mean, angles have common ray but the angles in a linear pair are not supplementary.
i. e., AOB, AOE; AOE, EOD and EOD, COD
Exercise 5.2
Question 1:
State the property that is used in each of the following statements:
(i) If a||b, then 1 = 5. (ii) If 4 = 6, then a||b.
(iii) If 4 + 5 + 180, then a||b.
SOLUTION 1:
(i) Given, a||b, then 1 = 5 [Corresponding angles]
If two parallel lines are cut by a transversal, each pair of corresponding angles are equal in measure.
(ii) Given, 4 = 6, then a||b [Alternate interior angles] When a transversal cuts two lines such that pairs of alternate interior angles are equal, the lines have to be parallel.
(iii) Given, 4 + 5 = 180, then a||b [Co-interior Angles]
When a transversal cuts two lines, such that pairs of interior angles on the same side of transversal are supplementary, the lines have to be parallel.
Question 2:
In the adjoining figure, identify:
(i) the pairs of corresponding angles.
(ii) the pairs of alternate interior angles.
(iii) the pairs of interior angles on the same side of the transversal. (iv) the vertically opposite angles.
SOLUTION 2:
(i) The pairs of corresponding angles:
1, 5; 2, 6; 4, 8 and 3, 7 (ii) The pairs of alternate interior angles are:
3, 5 and 2, 8
(iii) The pair of interior angles on the same side of the transversal:
3, 8 and 2, 5
(iv) The vertically opposite angles are:
1, 3; 2, 4; 6, 8 and 5, 7
Question 3:
In the adjoining figure, p||q. Find the unknown angles.
SOLUTION 3:
Given, p||q and cut by a transversal line.
125 e 180 [Linear pair]
e180125 55 ……….(i)
Now e f 55 [Vertically opposite angles]
Also a f 55 [Alternate interior angles]
a b 180 [Linear pair]
55 b 180 [From equation (i)]
b180 55 125
Now a c 55 and b d 125 [Vertically opposite angles]
Thus, a 55 ,b 125 ,c 55 ,d 125 ,e 55 and f 55 .
Question 4:
Find the values of x in each of the following figures if l||m
(i) Given, l||m and t is transversal line.
Interior vertically opposite angle between lines l and t 110 .
110 x 180 [Supplementary angles]
x 180110 70
(ii) Given, l||m and t is transversal line.
x2x 180 [Interior opposite angles]
3x 180
180
x 60
3
(iii) Given, l||m and a||b.
x 100 [Corresponding angles]
Question 5:
In the given figure, the arms of two angles are parallel. If ABC = 70 , then find:
(i) DGC (ii) DEF
SOLUTION 5:
(i) Given, AB || DE and BC is a transversal line and ABC 70
ABC = DGC [Corresponding angles]
DGC = 70 ……….(i)
(ii) Given, BC || EF and DE is a transversal line and DGC 70
DGC = DEF [Corresponding angles]
DEF = 70 [From equation (i)]
Question 6:
In the given figures below, decide whether l is parallel to m.
SOLUTION 6:
(i) 126 44 170
l ||m because sum of interior opposite angles should be 180 .
(ii) 75 75 150
l ||m because sum of angles does not obey the property of parallel lines.
(iii) 57123180 l ||m due to supplementary angles property of parallel lines.
(iv) 98 72 170
l is not parallel to m because sum of angles does not obey the property of parallel lines.
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