**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 90^{0}, then the two angles are said to be complementary.

Now,

a + a = 90^{o}

2a = 90^{o}

a = 90/2

x = 45^{0}

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 180^{0}, 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, 90^{0} + 90^{o} = 180^{o}

**8. An angle is greater than 45 ^{0}. 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.

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