NCERT Solutions for Class 7 Maths
Chapter 5 Lines and Angles
Exercise 5.1
1. Find the complement of each of the following angles:
Answer
(i) Complement angle of 20°=90° – 20°= 70°
(ii) Complement angle of 63°=90 – 63°= 27°
(iii) Complement angle of 57°= 90°- 57°= 33
2. Find the supplement of each of the following angles:
Answer
(i) Supplement angle of 105° = 180° – 105°= 75°
(ii) Supplement angle of 87 = 180° – 87° = 93°
(iii) Supplement angle of 154° = 180° – 154°= 26°
3. Identify which of the following pairs of angles are complementary and which are supplementary.
(i) 65º, 115º
(ii) 63º, 27º
(iii) 112º, 68º
(iv) 130º, 50º
(v) 45º, 45º
(vi) 80º, 10º
Answer
Firstly, find the sum of the given angles. If the sum of the angles is 90° then they are complementary and if the sum of the angles is 180° then they are supplementary.
∴ ∠x+ ∠y=65°+ 115°=180°
Hence, ∠x and ∠y are supplementary angles. (ii) Let ∠x=63° and ∠y= 27°
∴ ∠x+ ∠y= 63° + 27°= 90°
Hence, ∠x and ∠y are complementary angles. (iii) Let ∠x = 112° and y=68°
∴ ∠x+ ∠y=112° + 68° = 180°
Hence, ∠x and ∠y are supplementary angles.
(iv) Let ∠x= 130° and ∠y = 50
∴ ∠x+∠y = 130°+50° = 180°
Hence, ∠x and ∠y are supplementary angles.
(v) Let ∠x=45° and ∠y=45°
∴ ∠x+ ∠y = 45°+45° = 90
Hence, ∠x and ∠y are complementary angles.
(vi) Let ∠x=80° and ∠y= 10°
∴ ∠x+ ∠y = 80°+10° = 90°
Hence, ∠x and ∠y are complementary angles.
4. Find the angle which is equal to its complement.
Answer
Let the measure of the required angle be xo.
We know that, sum of measures of complementary angle pair is 90o.
Then,
= x + x = 90o
= 2x = 90o
= x = 90/2
= x = 45o
Hence, the required angle measures is 45o.
5. Find the angle which is equal to its supplement.
Answer
Let the measure of the required angle be xo.
We know that, sum of measures of supplementary angle pair is 180o.
Then,
= x + x = 180o
= 2x = 180o
= x = 180/2
= x = 90o
Hence, the required angle measures is 90o.
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.
Answer
From the question, it is given that,
∠1 and ∠2 are supplementary angles.
If ∠1 is decreased, then ∠2 must be increased by the same value.
Hence, this angle pair remains supplementary.
7. Can two angles be supplementary if both of them are:
(i) acute? (ii) obtuse? (iii) right?
Answer
(i) No. If two angles are acute, means less than 90o, the two angles cannot be supplementary.
Because, their sum will be always less than 90o.
(ii) No. If two angles are obtuse, means more than 90o, the two angles cannot be supplementary.
Because, their sum will be always more than 180o.
(iii) Yes. If two angles are right, means both measures 90o, then two angles can form a supplementary pair.
∴ 90o + 90o = 180
8. An angle is greater than 45º. Is its complementary angle greater than 45º or equal to 45º or less than 45º?
Answer
Let us assume the complementary angles be p and q,
We know that, sum of measures of complementary angle pair is 90o.
Then,
= p + q = 90o
It is given in the question that p > 45o
Adding q on both the sides,
= p + q > 45o + q
= 90o > 45o + q
= 90o – 45o > q
= q < 45o
Hence, its complementary angle is less than 45o.
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?
Answer
(i) By observing the figure we came to conclude that,
Yes, as ∠1 and ∠2 having a common vertex i.e. O and a common arm OC.
Their non-common arms OA and OE are on both the side of common arm.
(ii) By observing the figure, we came to conclude that,
No, since they are having a common vertex O and common arm OA.
But, they have no non-common arms on both the side of the common arm.
(iii) By observing the figure, we came to conclude that,
Yes, as ∠COE and ∠EOD having a common vertex i.e. O and a common arm OE.
Their non-common arms OC and OD are on both the side of common arm.
(iv) By observing the figure, we came to conclude that,
Yes, as ∠BOD and ∠DOA having a common vertex i.e. O and a common arm OE.
Their non-common arms OA and OB are opposite to each other.
(v) Yes, ∠1 and ∠2 are formed by the intersection of two straight lines AB and CD.
(vi) ∠COB is the vertically opposite angle of ∠5. Because these two angles are formed by the intersection of two straight lines AB and CD.
10. Indicate which pairs of angles are:
(i) Vertically opposite angles. (ii) Linear pairs.
Answer
∠1 and ∠4, ∠5 and ∠2 + ∠3 are vertically opposite angles. Because these two angles are formed by the intersection of two straight lines. (ii) By observing the figure we can say that,
∠1 and ∠5, ∠5 and ∠4 as these are having a common vertex and also having non common arms opposite to each other.11. In the following figure, is ∠1 adjacent to ∠2? Give reasons.
Answer
∠1 and ∠2 are not adjacent angles. Because, they are not lie on the same vertex.
12. Find the values of the angles x, y, and z in each of the following:
Answer
(i) ∠x = 55o, because vertically opposite angles.
∠x + ∠y = 180o … [∵ linear pair]
= 55o + ∠y = 180o
= ∠y = 180o – 55o
= ∠y = 125o
Then, ∠y = ∠z … [∵ vertically opposite angles]
∴ ∠z = 125o
(ii) ∠z = 40o, because vertically opposite angles.
∠y + ∠z = 180o … [∵ linear pair]
= ∠y + 40o = 180o
= ∠y = 180o – 40o
= ∠y = 140o
Then, 40 + ∠x + 25 = 180o … [∵angles on straight line]
65 + ∠x = 180o
∠x = 180o – 65
∴ ∠x = 115o
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 at 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 __________.
Answer
(i) If two angles. are complementary, then the sum of their measures is 90°
(ii) If two angles are supplementary, then the sum of their measures is 180°
(iii) Two angles forming a linear pair are adjacent angles or supplementary.
(iv) If two adjacent angles are supplementary, then they form a linear pair.
(v) If two lines intersect at a point, then the vertically opposite angles are always equal.
(vi) If two lines intersect at a point and one pair of vertically opposite angles are acute angles, then the other pair of vertically opposite angles are obtuse angles.
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
Answer
(i) A pair of obtuse vertically opposite angles are ∠AOD and ∠BOC.
(ii) Adjacent complementary angles are ∠AOB and ∠AOE.
(iii) Equal supplementary angles are ∠BOE and ∠EOD.
(iv) Unequal supplementary angles are ∠E0A and ∠E0C.
(v) Adjacent angles that do not form a linear pair are
∠AOB, ∠AOE, ∠AOE, ∠EOD, ∠EOD, ∠COD