**NCERT Solutions for Class 7 Maths**

Chapter 5 Lines and Angles

Exercise 5.1

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

(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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