## Exercise 5.1

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

Solution:

(i) Complement of 20° = 90° – 20° = 70°

(ii) Complement of 63° = 90° – 63° = 27°

(iii) Complement of 57° = 90° – 57° = 33°

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

Solution:

(i) Supplement of 105° = 180° – 105° = 75°

(ii) Supplement of 87° = 180° – 87° = 93°

(iii) Supplement of 154° = 180° – 154° = 26°

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

**(i) 65 ^{o}, 115^{o}**

**Solution:-**

We have to find the sum of given angles to identify whether the angles are complementary or supplementary.

Then,

= 65^{o} + 115^{o}

= 180^{o}

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

∴These angles are supplementary angles.

**(ii) 63 ^{o}, 27^{o}**

**Solution:-**

We have to find the sum of given angles to identify whether the angles are complementary or supplementary.

Then,

= 63^{o} + 27^{o}

= 90^{o}

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

∴These angles are complementary angles.

**(iii) 112 ^{o}, 68^{o}**

**Solution:-**

We have to find the sum of given angles to identify whether the angles are complementary or supplementary.

Then,

= 112^{o} + 68^{o}

= 180^{o}

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

∴These angles are supplementary angles.

**(iv) 130 ^{o}, 50^{o}**

**Solution:-**

Then,

= 130^{o} + 50^{o}

= 180^{o}

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

∴These angles are supplementary angles.

**(v) 45 ^{o}, 45^{o}**

**Solution:-**

Then,

= 45^{o} + 45^{o}

= 90^{o}

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

∴These angles are complementary angles.

**(vi) 80 ^{o}, 10^{o}**

**Solution:-**

Then,

= 80^{o} + 10^{o}

= 90^{o}

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

∴These angles are complementary angles.

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

**Solution:-**

Let the measure of the required angle be x^{o}.

We know that, sum of measures of complementary angle pair is 90^{o}.

Then,

= x + x = 90^{o}

= 2x = 90^{o}

= x = 90/2

= x = 45^{o}

Hence, the required angle measures is 45^{o}.

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

**Solution:-**

Let the measure of the required angle be x^{o}.

We know that, sum of measures of supplementary angle pair is 180^{o}.

Then,

= x + x = 180^{o}

= 2x = 180^{o}

= x = 180/2

= x = 90^{o}

Hence, the required angle measures 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:-**

From the question, it is given that,

∠1 and ∠2 are supplementary angles.

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. If two angles are acute, means less than 90^{o}, the two angles cannot be supplementary. Because, their sum will be always less than 90^{o}.

**(ii). Obtuse?**

**Solution:-**

No. If two angles are obtuse, means more than 90^{o}, the two angles cannot be supplementary. Because, their sum will be always more than 180^{o}.

**(iii). Right?**

**Solution:-**

Yes. If two angles are right, means both measures 90^{o}, then two angles can form a supplementary pair.

∴90^{o }+ 90^{o} = 180

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

**Solution:-**

Let us assume the complementary angles be p and q,

We know that, sum of measures of complementary angle pair is 90^{o}.

Then,

= p + q = 90^{o}

It is given in the question that p > 45^{o}

Adding q on both the sides,

= p + q > 45^{o }+ q

= 90^{o} > 45^{o }+ q

= 90^{o} – 45^{o} > q

= q < 45^{o}

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

**9. In the adjoining figure:**

**i) Is ∠1 adjacent to ∠2?**

**Solution:-**

By observing the figure we came to conclude that,

Yes, as ∠1 and ∠2 having a common vertex i.e. O and a common arm OC.

Their non-common arms OA and OE are on both the side of common arm.

**(ii) Is ∠AOC adjacent to ∠AOE?**

**Solution:-**

By observing the figure, we came to conclude that,

No, since they are having a common vertex O and common arm OA.

But, they have no non-common arms on both the side of the common arm.

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

**Solution:-**

By observing the figure, we came to conclude that,

Yes, as ∠COE and ∠EOD having a common vertex i.e. O and a common arm OE.

Their non-common arms OC and OD are on both the side of common arm.

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

**Solution:-**

By observing the figure, we came to conclude that,

Yes, as ∠BOD and ∠DOA having 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:-**

Yes, ∠1 and ∠2 are formed by the intersection of two straight lines AB and CD.

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

**Solution:-**

∠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:-**

By observing the 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:-**

By observing the figure we can say that,

∠1 and ∠5, ∠5 and ∠4 as these are having a common vertex and also having non common arms opposite to each other.

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

**Solution:-**

∠1 and ∠2 are not adjacent angles. Because, they are not lie on the same vertex.

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

**Solution:- (i)**

∠x = 55^{o}, because vertically opposite angles.

∠x + ∠y = 180^{o} … [∵ linear pair]

= 55^{o} + ∠y = 180^{o}

= ∠y = 180^{o} – 55^{o}

= ∠y = 125^{o}

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

∴ ∠z = 125^{o}

**(ii)**

**Solution:-**

∠z = 40^{o}, because vertically opposite angles.

∠y + ∠z = 180^{o} … [∵ linear pair]

= ∠y + 40^{o} = 180^{o}

= ∠y = 180^{o} – 40^{o}

= ∠y = 140^{o}

Then, 40 + ∠x + 25 = 180^{o} … [∵angles on straight line]

65 + ∠x = 180^{o}

∠x = 180^{o} – 65

∴ ∠x = 115^{o}

**13. Fill in the blanks:**

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 ______ .

Solution:

(i) 90°

(ii) 180°

(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**

**Solution:-**

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

**(ii) Adjacent complementary angles**

**Solution:-**

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

**(iii) Equal supplementary angles**

**Solution:-**

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

**(iv) Unequal supplementary angles**

**Solution:-**

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

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

**Solution:-**

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