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.
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
We know that, two angles are complementary, if their sum is $90^{\circ}$.
Let the angle be $x^{\circ}$.
Therefore, its complement be $90^{\circ}-x^{\circ}$.
Since, the angle is equal to its complement.
$x^{0}=90^{\circ}-x^{\circ}$
On transposing $x^{\circ}$ from RHS to LHS, we get
$x^{\circ}+x^{\circ}=90^{\circ} \Rightarrow 2 x^{\circ}=90^{\circ}$
On dividing both sides by 2, we get
$\frac{2 x}{2}=\frac{90^{\circ}}{2} \quad \Rightarrow \quad x=45^{\circ}$
Hence, the required angle is $45^{\circ}$.
5. Find the angle which is equal to its supplement.
Answer
We know that, two angles are supplementary, if their sum is $180^{\circ}$.
Let the angle be $x^{\circ}$.
Therefore, its supplement be $180^{\circ}-x^{\circ}$.
Since, the angle is equal to it supplement.
$\therefore \quad x^{\circ}=180^{\circ}-x^{\
On transposing $x^{\circ}$ from RHS to LHS, we get
$x^{\circ}+x^{\circ}=180^{\
On dividing both by 2 , we get
$x^{\circ}=\frac{180^{\circ}}{
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
We know that, the two angles are supplementary, if their sum is $180^{\circ}$.
$\angle 1+\angle 2=180^{\circ}$ [given]
$\therefore$ Now, if $\angle 1$ is decreased, then $\angle 2$ should be increased, so that both the angles still remain supplementary.
i.e. $\angle 1+\angle 2=180^{\circ}$
7. Can two angles be supplementary if both of them are:
(i) acute? (ii) obtuse? (iii) right?
Answer
(i) No, because sum of two acute angle is always less than $180^{\circ}$. So, two acute angles cannot be supplementary.
(ii) No, because sum of two obtuse angles is always greater than $180^{\circ}$.
So, two obtuse angles can never be supplementary.
(iii) Yes, because sum of two right angles is always equal to $180^{\circ}$. So, two right angles are supplementary,
8. An angle is greater than 45º. Is its complementary angle greater than 45º or equal to 45º or less than 45º?
Answer
We know that, the sum of two complementary angles is $90^{\circ}$, So, if an knele is greater than $45^{\circ}$, then its complementary angle would be less then $45^{\circ}$.
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) Yes, $\angle 1$ is adjacent to $\angle 2$ because $\angle 1$ and $\angle 2$ have common vertex $O$. Also, they have a common arm $O C$ and their other arms $O A$ and $O E$ lie on the opposite side of the common arm $O C$.
(ii) No, $\angle A O C$ is not adjacent to $\angle A O E$ because $\angle A O C$ is a part of $\angle A O E$ i.e. they have common interior point $C$.
(iii) Yes, $\angle C O E$ and $\angle E O D$ form a linear pair because $C O D$ is a straight line.
(iv) Yes, $\angle B O D$ and $\angle D O A$ are supplementary because $\angle B O D+\angle D O A=180^{\circ}$.
(v) Yes, $\angle 1$ is vertically opposite to $\angle 4$.
(vi) Vertically opposite angle of $\angle 5$ is $\angle B O C$
$=\angle C O E+\angle B O E=\angle 2+\angle 3$
10. Indicate which pairs of angles are:
(i) Vertically opposite angles. (ii) Linear pairs.
Answer
The vertically opposite angles are $\angle 1, \angle 4$ and $\angle 5, \angle 2+\angle 3$(ii) A linear pair is a pair of adjacent angles whose non-common sides are opposite rays.
Therefore, $\angle 1, \angle 5$ and $\angle 4, \angle 5$ are form a linear pair.
11. In the following figure, is ∠1 adjacent to ∠2? Give reasons.
Answer
No, $\angle 1$ is not adjacent to $\angle 2$ because $\angle 1$ and $\angle 2$ do not have common vertex.
12. Find the values of the angles x, y, and z in each of the following:
Answer
(i) $\quad \angle x=55^{\circ} \quad$ [vertically opposite angles]
$\angle y+55^{\circ}=180^{\circ} \quad$ [by linear pair]
$\therefore \quad \angle y=180^{\circ}-55^{\circ}=125^{
$\angle z=\angle y=125^{\circ} \quad$ [vertically opposite angles]
Hence, $\angle x=55^{\circ}, \angle y=125^{\circ}$, and $\angle z=125^{\circ}$
$\begin{array}{ll}\text { (ii) } \angle y+40^{\circ}=180^{\circ} & \text { [by linear pair] }\end{array}$
$\Rightarrow \quad \angle y=180^{\circ}-40^{\circ} \Rightarrow \angle y=140^{\circ}$
$\begin{aligned} \angle z=40^{\circ} & \text { [vertically opposite angles] } \end{aligned}$
$\because \quad \angle x+25^{\circ}+\angle z=180^{\circ}$ [by linear pair]
$\therefore \quad \angle x+25^{\circ}+40^{\circ}=180^{\
Hence, $\angle x=115^{\circ}, \angle y=140^{\circ}$ and $\angle z=40^{\circ}$
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
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