EXERCISE 5.1

**1) Find the complement of each of the following angles:
Ans:
**

i)The given angle is 20ᵒ

Let its complement be x

Therefore, x + 20ᵒ = 90ᵒ

x = 90ᵒ − 20ᵒ

x = 70ᵒ

ii)The given angle is 63ᵒ

Let its complement be x

Therefore, x + 63ᵒ = 90ᵒ

x = 90ᵒ − 63ᵒ

x = 27ᵒ

iii)The given angle is 57ᵒ

Let its complement be x

Therefore, x + 57ᵒ = 90ᵒ

x = 90ᵒ − 57ᵒ

x = 33ᵒ

**2) Find the supplement of each of the following angles:
Ans:
**

i)The given angle is 105ᵒ

Let its supplement be xᵒ

x + 105ᵒ = 180ᵒ → x = 180ᵒ − 105ᵒ → x = 75ᵒ

ii)The given angle is 87ᵒ

Let its supplement be xᵒ

x + 87ᵒ = 180ᵒ → x = 180ᵒ − 87ᵒ → x = 93ᵒ

iii)The given angle is 154ᵒ

Let the supplement be xᵒ

x + 154ᵒ = 180ᵒ → x = 180ᵒ − 154ᵒ → x = 26ᵒ

**3). Identify which of the following pairs of angles are complementary and which are supplementary :**

**65ᵒ , 115ᵒ**

**Ans:**

i) 65ᵒ + 115ᵒ

= 180ᵒ

Therefore, these angles are supplementary.

ii) 63ᵒ , 27ᵒ

Ans: 63ᵒ + 27ᵒ

= 90ᵒ

Therefore, these angles are complementary.

iii)112ᵒ, 68ᵒ

Ans: 112ᵒ + 68ᵒ

= 180ᵒ

Therefore, the angles are supplementary.

iv) 130ᵒ , 50ᵒ

Ans: 130ᵒ + 50ᵒ

= 180ᵒ

Therefore, these angles are supplementary.

v) 45ᵒ, 45ᵒ

Ans: : 45ᵒ + 45ᵒ = 90ᵒ

Therefore, these angles are complementary.

**4) Find the angle which is equal to its complement.**

**Ans:** Let the measure of the required angle be x.

Since the sum of the complementary angle is 90ᵒ.

x + x = 90ᵒ

2 x = 90ᵒ

x = 90ᵒ/2

x = 45ᵒ

Hence the required angle measure is 45ᵒ.

**5)Find the angle which is equal to its supplement.**

**Ans:** Let the measure of the required angle be x

We know that sum of measure of supplementary angle is 180ᵒ.

Therefore, x + x = 180ᵒ

2 x = 180ᵒ

x = 180ᵒ/2

x = 90ᵒ

Hence, the required angle measure is 90ᵒ.

**6)In the given, ∠1 and ∠2 are supplementary. If ∠1 is decreased, what changes should take place in ∠2 so that both the angles still remain supplementary**

**Ans: **

It is given that ∠1 and ∠2 are supplementary

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 both them are:**

**i) Acute?**

**Ans:** No. If two angles are acute, they cannot be supplementary as each angle is less than 90ᵒ and hence their sum wont be 180ᵒ.

**ii) Obtuse?**

**Ans:** No. If two angles are obtuse, then each angle measure is more than 90ᵒ and so their sum will be always more than 180ᵒ.

**iii) Right?**

**Ans:** Yes. If two angles are right, it means that their sum will be 180ᵒ as measure of each right angle is 90ᵒ.

**8) An angle is greater than 45ᵒ. Is its complementary angle greater than 45ᵒ or equal to 45ᵒ or less than 45ᵒ?**

**Ans:** Let us assume the complementary angles to be a and b

We know that the sum of measures of complementary angle pair is 90ᵒ

Then a + b = 90ᵒ

It is given that a > 45ᵒ

Adding b on both sides

= a + b > 45ᵒ + b

= 90ᵒ > 45ᵒ + b

= 90ᵒ − 45ᵒ > b

= b < 45ᵒ

Hence, its complementary angle is less than 45ᵒ.

**9) In the adjoining figure:
**

**i) Is ∠1 adjacent to ∠2?**

**Ans:**Yes.

**ii) Is ∠AOC adjacent to ∠AOE?**

**Ans:**No. 0

**iii) Do ∠COE and ∠CEO form a linear pair?**

**Ans:**Yes.

**iv) Are ∠BOD and ∠DOA supplementary?**

**Ans:**Yes.

**v) Is ∠1 vertically opposite to ∠4?**

**Ans:**Yes.

**vi) What is the vertically opposite angle of ∠5?**

**Ans:**∠COB.

**10) Indicate which pairs of angles are:
**

**(i) Vertically opposite angles**

**Ans:**∠1 and ∠4 , ∠5 and ∠2+ ∠

**(ii) Linear pairs**

**Ans:**∠1 and ∠5 and ∠5 and ∠4

**11) In the following figure, is ∠1 adjacent to ∠2 ? Give reasons.
**

**Ans:**No, ∠1 and ∠2 are not adjacent angles because they do not have a

common vertex.

**12) Find the values of the angles x, y and z in each of the following:**

**Ans:
**

i) ∠y + 55ᵒ = 180ᵒ —- Linear pair

∠y = 180ᵒ − 55ᵒ

∠y = 125ᵒ

∠x = 55ᵒ —– vertically opposite angles

∠y = ∠x —— vertically opposite angles

.’ . ∠z = 125ᵒ

ii) ∠40ᵒ + ∠x + 25ᵒ = 180ᵒ —- angles on a straight line

∠x = 180ᵒ − 65ᵒ

∠x = 115ᵒ

∠z = 40ᵒ —— vertically opposite angles

∠y + ∠z = 180ᵒ

∠y = 180ᵒ − ∠z

= 180ᵒ − 40ᵒ

= 140ᵒ

**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 ________.**

**Ans:**

i) 90ᵒ

ii) 180ᵒ

iii) Supplementary

iv) Linear pair

v) Equal

vi) Obtuse angle

**14) In the adjoining figure, name the following pairs of angles:
**

**i) Obtuse vertically opposite angles**

**Ans:**∠AOD and ∠BOC

**ii) Adjacent complementary angles**

**Ans:**∠EOA and ∠AOB

**iii) Equal supplementary angles**

**Ans:**∠EOC and ∠EOA

**iv) Unequal supplementary angles**

**Ans:**∠EOC and ∠EOA

**v) Adjacent angles that do not form a linear pair**

**Ans:**∠EOD and ∠DOC , ∠BOA and ∠AOE , ∠AOE and ∠DOE