1. Find the complement of each of the following angles:

We know that two angles are said to be complementary if the sum of their measures is 90^o.

(i)

Solution:

Given angle is 20^o

Let the measure of its complement be a^o.

Then,

a + 20^o = 90^o

a = 90^o – 20^o

 a = 70^o

Hence, the complement of the given angle measures 70^o.

(ii)

Solution:

 Given angle is 63^o

Let the measure of its complement be a^o.

Then,

 a+ 63^o = 90^o

 a = 90^o – 63^o

 a = 27^o

Hence, the complement of the given angle measures 27^o.

(iii)

Solution:-

Given angle is 57^o

Let the measure of its complement be a^o.

Then,

a + 57^o = 90^o

a = 90^o – 57^o

a = 33^o

Hence, the complement of the given angle measures 33^o.

2. Find the supplement of each of the following angles:

We know that two angles are said to be supplementary if the sum of their measures is 180^o.

(i)

Solution:

 Given angle is 105^o

Let the measure of its supplement be a^o.

Then,

a + 105^o = 180^o

a = 180^o – 105^o

a = 75^o

Hence, the supplement of the given angle measures 75^o.

(ii)

Solution:

Given angle is 87^o

Let the measure of its supplement be a^o.

Then,

a + 87^o = 180^o

a = 180o – 87^o

a = 93^o

Hence, the supplement of the given angle measures 93^o.

(iii)

Solution:

Given angle is 154^o

Let the measure of its supplement be a^o.

Then,

a + 154^o = 180^o

a = 180^o – 154^o

a = 26^o

Hence, the supplement of the given angle measures 26^o.

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

->If the sum of two angle measures is 180 ^o, then the two angles are said to be supplementary.

->If the sum of two angle measures is 90^o, then the two angles are said to be complementary.

(i) 65^o, 115^o

Solution:

We have to find the sum of given angles is complementary or supplementary.

Then,

= 65^o + 115^o

= 180^o

Hence, These angles are supplementary angles.

(ii) 63^o, 27^o

Solution:

We have to find the sum of given angles is complementary or supplementary.

Then,

= 63^o + 27^o

= 90^o

Hence, These angles are complementary angles.

(iii) 112^o, 68^o

Solution:

We have to find the sum of given angles is complementary or supplementary.

Then,

= 112^o + 68^o

= 180^o

Hence, These angles are supplementary angles.

(iv) 130^o, 50^o

Solution:

We have to find the sum of given angles is complementary or supplementary.

Then,

= 130^o + 50^o

= 180^o

Hence, These angles are supplementary angles.

(v) 45^o, 45^o

Solution:

We have to find the sum of given angles is complementary or supplementary.

Then,

= 45^o + 45^o

= 90^o

Hence, These angles are complementary angles.

(vi) 80^o, 10^o

Solution:

We have to find the sum of given angles is complementary or supplementary.

Then,

= 80^o + 10^o

= 90^o

Hence, These angles are complementary angles.

4. Find the angles which is equal to its complement.

Solution:

Let the measure of the required angle be a^o.

We know that the sum of two angle measures is 90⁰, then the two angles are said to be complementary.

Now,

a + a = 90^o

2a = 90^o

a = 90/2

x = 45⁰

Hence, the measure of required angle is 45^o.

5. Find the angles which is equal to its supplement.

Solution:

Let the measure of the required angle be a^o.

We know that the sum of two angle measures is 180⁰, then the two angles are said to be Supplementary.

Now,

a + a = 180^o

2a = 180^o

a = 180/2

a = 90^o

Hence, the measures of required angle is 90^o.

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 angles still remain supplementary?

Solution:

Given,

∠1 and ∠2 are supplementary angles.

Now,

As we know that the angles are supplementary so, 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?

Solution:

No, the two acute angles are never supplementary because the acute angle is less than 90^o and their sum will be always less than 90^o. That’s why the two angles cannot be supplementary.

(ii). Obtuse?

Solution:

No, the two obtuse angles are never supplementary because the obtuse angle means more than 90^o, and their sum will be always more than 180^o. That’s why the two obtuse angles cannot be supplementary.

(iii). Right?

Solution:

Yes, the two right angles are supplementary because the angles are right, which means both measure 90o, and their sum always is 180^o.

Hence, 90⁰ + 90^o = 180^o

8. An angle is greater than 45⁰. Is its complementary angle greater than 45^o or equal to 45^o or less than 45^o?

Solution:

Let the complementary angles be x and y,

We know that the sum of two complementary angles is 90^o.

Then,

 x + y = 90^o

 According to question

Let given angle, x > 45^o

Adding y , on both the sides,

 x + y > 45^o + y

 90^o > 45^o + y

 90^o – 45^o > y

 y <45^o

Hence, its complementary angle is less than 45^o.

9. In the adjoining figure:

(i) Is ∠1 adjacent to ∠2?

Solution:

Yes,  ∠1 and ∠2 are adjacent angles because they have a common vertex i.e. O, and a common arm OC. They have also non-common arms i.e. OA AND OE that are on both sides of common arms.

(ii) Is ∠AOC adjacent to ∠AOE?

Solution:

No, ∠AOC and ∠AOE are not adjacent. They are having a common vertex O and common arm OA but, they have no non-common arms on both sides of the common arm.

(iii) Do ∠COE and ∠EOD form a linear pair?

Solution:

Yes, ∠COE and ∠EOD are adjacent angles because they have a common vertex i.e. O, and a common arm OE. Their non-common arms OC and OD are on both sides of the common arm.

(iv) Are ∠BOD and ∠DOA supplementary?

Solution:

Yes, ∠BOD and ∠DOA are adjacent because they have a common vertex i.e. O, and a common arm OE. Their non-common arms OA and OB are opposite to each other.

(v) Is ∠1 vertically opposite to ∠4?

Solution:

By the definition of the vertically opposite angle, we know that the two angles are vertically opposite when they are formed by the intersection of two straight angles.

Yes, ∠1 and ∠2 are vertically opposite angles because they are formed by the intersection of two straight lines AB and CD.

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

Solution:

By the definition of the vertically opposite angle, we know that the two angles are vertically opposite when they are formed by the intersection of two straight angles.

Hence, ∠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.

Solution:

According to the given figure, we can say that,

∠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) Linear pairs.

Solution:

According to the given figure, we can say that,

∠1 and ∠5, ∠5, and ∠4 are in linear pairs because these are having a common vertex and also have non-common arms opposite to each other.

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

Solution:

By observing the given figure,

∠1 and ∠2 are not adjacent angles because they are not lying on the same vertex.

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

(i)

Solution:

By observing the given figure, ∠x = 55^o, (because vertically opposite angles)

∠x + ∠y = 180^o [because of linear pair]

 55^o + ∠y = 180^o

 ∠y = 180^o – 55^o

 ∠y = 125^o

Then, ∠y = ∠z [ vertically opposite angles]

Hence, ∠z = 125^o

(ii)

Solution:

 By observing the given figure, ∠z = 40^o (because of vertically opposite angles)

∠y + ∠z = 180^o [because of linear pair]

∠y + 40^o = 180^o

 ∠y = 180^o – 40^o

∠y = 140^o

Now,

40 + ∠x + 25 = 180^o [because angles on straight line]

65 + ∠x = 180^o

∠x = 180^o – 65

Hence, ∠x = 115^o

13. Fill in the blanks:

(i) If two angles are complementary, then the sum of their measures is _______.

Solution:-

If two angles are complementary, then the sum of their measures is 90^o.

(ii) If two angles are supplementary, then the sum of their measures is ______.

Solution:

If two angles are supplementary, then the sum of their measures is 180^o.

(iii) Two angles forming a linear pair are _______________.

Solution:

Two angles forming a linear pair are Supplementary.

(iv) If two adjacent angles are supplementary, they form a ___________.

Solution:

If two adjacent angles are supplementary, they form a linear pair.

(v) If two lines intersect at a point, then the vertically opposite angles are always_____________.

Solution:

If two lines intersect at a point, then the vertically opposite angles are always equal.

(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:

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 Obtuse angles.

14. In the adjoining figure, name the following pairs of angles.

(i) Obtuse vertically opposite angles

Solution:

in the given figure ∠AOD and ∠BOC are obtuse vertically opposite angles.

(ii) Adjacent complementary angles

Solution:

in the given figure ∠EOA and ∠AOB are adjacent complementary angles.

(iii) Equal supplementary angles

Solution:

in the given figure ∠EOB and EOD are the equal supplementary angles.

(iv) Unequal supplementary angles

Solution:

in the given figure ∠EOA and ∠EOC are the unequal supplementary angles.

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

Solution:

in the given figure ∠AOB and ∠AOE, ∠AOE and ∠EOD, ∠EOD and ∠COD are the adjacent angles that do not form a linear pair.