Chapter 5: Lines and Angles

Solution:-

Two angles are said to be complementary if the sum of their measures is 90^o.

The given angle is 20^o

Let the measure of its complement be x^o.

Then,

= x + 20^o = 90^o

= x = 90^o – 20^o

= x = 70^o

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

(ii)

Solution:-

Two angles are said to be complementary if the sum of their measures is 90^o.

The given angle is 63^o

Let the measure of its complement be x^o.

Then,

= x + 63^o = 90^o

= x = 90^o – 63^o

= x = 27^o

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

(iii)

Solution:-

Two angles are said to be complementary if the sum of their measures is 90^o.

The given angle is 57^o

Let the measure of its complement be x^o.

Then,

= x + 57^o = 90^o

= x = 90^o – 57^o

= x = 33^o

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

Solution:-

Two angles are said to be supplementary if the sum of their measures is 180^o.

The given angle is 105^o

Let the measure of its supplement be x^o.

Then,

= x + 105^o = 180^o

= x = 180^o – 105^o

= x = 75^o

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

(ii)

Solution:-

Two angles are said to be supplementary if the sum of their measures is 180^o.

The given angle is 87^o

Let the measure of its supplement be x^o.

Then,

= x + 87^o = 180^o

= x = 180^o – 87^o

= x = 93^o

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

(iii)

Solution:-

Two angles are said to be supplementary if the sum of their measures is 180^o.

The given angle is 154^o

Let the measure of its supplement be x^o.

Then,

= x + 154^o = 180^o

= x = 180^o – 154^o

= x = 26^o

Hence, the supplement of the given angle measures 93^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:-

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

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:-

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

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:-

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

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.

Solution:-

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

We know that the 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 measure is 45^o.

Solution:-

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

We know that the 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 measure is 90^o.

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.

Solution:-

No. If two angles are acute, which means less than 90^o, then they cannot be supplementary because their sum will always be less than 90^o.

(ii). Obtuse?

Solution:-

No. If two angles are obtuse, which means more than 90^o, then they cannot be supplementary because their sum will always be more than 180^o.

(iii). Right?

Solution:-

Yes. If two angles are right, which means both measure 90^o, then they can form a supplementary pair.

∴ 90^o + 90^o = 180

Solution:-

Let us assume the complementary angles be p and q,

We know that the 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 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.

Solution:-

By observing the figure, we came to conclude that,

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

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

(ii) Is ∠AOC adjacent to ∠AOE?

Solution:-

By observing the figure, we came to conclude that,

No, since they have 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:-

By observing the figure, we came to conclude that,

Yes, as ∠COE and ∠EOD 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:-

By observing the figure, we came to conclude that,

Yes, as ∠BOD and ∠DOA 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:-

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.

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 have a common vertex and also have non-common arms opposite to each other.