Class 7 Lines and Angles Ex 5.1

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

(i)



Solution:-

Two angles are said to be complementary if the sum of their measures is \(90^{\circ}\).

The given angle is \(20^{\circ}\)

Let the measure of its complement be \(x^{\circ}\).

\(x + 20^{\circ} = 90^{\circ}\)

\(x = 90^{\circ} – 20^{\circ}\)

\(x = 70^{\circ}\)

Hence, the complement of the given angle measures \(70^{\circ}\).

(ii)



Solution:-

Two angles are said to be complementary if the sum of their measures is \(90^{\circ}\).

The given angle is \(63^{\circ}\)

Let the measure of its complement be \(x^{\circ}\).

\(x + 63^{\circ} = 90^{\circ}\)

\(x = 90^{\circ} – 63^{\circ}\)

\(x = 27^{\circ}\)

Hence, the complement of the given angle measures \(27^{\circ}\).

(iii)



Solution:-

Two angles are said to be complementary if the sum of their measures is \(90^{\circ}\).

The given angle is \(57^{\circ}\)

Let the measure of its complement be \(x^{\circ}\).

\(x + 57^{\circ} = 90^{\circ}\)

\(x = 90^{\circ} – 57^{\circ}\)

\(x = 33^{\circ}\)

Hence, the complement of the given angle measures \(33^{\circ}\).

2. Find the supplement of each of the following angles:
(i)


Solution:-

Two angles are said to be supplementary if the sum of their measures is \(180^{\circ}\).

The given angle is \(105^{\circ}\)

Let the measure of its supplement be \(x^{\circ}\).

\(x + 105^{\circ} = 180^{\circ}\)

\(x = 180^{\circ} – 105^{\circ}\)

\(x = 75^{\circ}\)

Hence, the supplement of the given angle measures \(75^{\circ}\).

(ii)


Solution:-

Two angles are said to be supplementary if the sum of their measures is \(180^{\circ}\).

The given angle is \(87^{\circ}\)

Let the measure of its supplement be \(x^{\circ}\).

\(x + 87^{\circ} = 180^{\circ}\)

\(x = 180^{\circ} – 87^{\circ}\)

\(x = 93^{\circ}\)

Hence, the supplement of the given angle measures \(93^{\circ}\).

(iii)



Solution:-

Two angles are said to be supplementary if the sum of their measures is \(180^{\circ}\).

The given angle is \(154^{\circ}\)

Let the measure of its supplement be \(x^{\circ}\)

\(x + 154^{\circ} = 180^{\circ}\)

\(x = 180^{\circ} – 154^{\circ}\)

\(x = 26^{\circ}\)

Hence, the supplement of the given angle measures \(93^{\circ}\).

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

(i) \(65^{\circ}, 115^{\circ}\)

Solution:-

\(65^{\circ} + 115^{\circ}= 180^{\circ}\)
If the sum of two angle measures is \(180^{\circ}\), then the two angles are said to be supplementary.
∴ These angles are supplementary angles.

(ii) \(63^{\circ}, 27^{\circ}\)
Solution:-

\(63^{\circ} + 27^{\circ}= 90^{\circ}\)
If the sum of two angle measures is \(90^{\circ}\), then the two angles are said to be complementary.
∴ These angles are complementary angles.

(iii) \(112^{\circ}, 68^{\circ}\)
Solution:

\(112^{\circ} + 68^{\circ} = 180^{\circ}\)
If the sum of two angle measures is \(180^{\circ}\), then the two angles are said to be supplementary.
∴ These angles are supplementary angles.

(iv) \(130^{\circ}, 50^{\circ}\)
Solution:-

\(130^{\circ} + 50^{\circ} = 180^{\circ}\)
If the sum of two angle measures is \(180^{\circ}\), then the two angles are said to be supplementary.
∴ These angles are supplementary angles.

(v) \(45^{\circ}, 45^{\circ}\)
Solution:-

\(45^{\circ} + 45^{\circ} = 90^{\circ}\)
If the sum of two angle measures is \(90^{\circ}\), then the two angles are said to be complementary.
∴ These angles are complementary angles.

(vi) \(80^{\circ}, 10^{\circ}\)
Solution:-
\(80^{\circ} + 10^{\circ} = 90^{\circ}\).
If the sum of two angle measures is \(90^{\circ}\)., then the two angles are said to be complementary.
∴ These angles are complementary angles.

4. Find the angles which are equal to their complement.
Solution:-

Let the measure of the required angle be \(x^{\circ}\).
Sum of measures of complementary angle pair is 90^{\circ}\)..
\( x + x = 90^{\circ}\).
\(2x = 90^{\circ}\).
\(x = \dfrac{90^{\circ}}{ 2}\)
\( x = 45^{\circ}\).
Hence, the required angle measure is \(45^{\circ}\).

5. Find the angles which are equal to their supplement.

Solution:-

Let the measure of the required angle be \({x^{\circ}}\).

We know that the sum of measures of supplementary angle pair is \({180^{\circ}}\).

\(x + x = \dfrac{180^{\circ}}{ 2}\)

\(2x = \dfrac{180^{\circ}}{ 2}\)

\(x = \dfrac{180^{\circ}}{ 2}\)

\(x = 90^{\circ}\)

Hence, the required angle measure is \(90^{\circ}\).

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:-

∠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, which means less than \(90^{\circ}\), then they cannot be supplementary because their sum will always be less than \(90^{\circ}\).

(ii). Obtuse?

Solution:-

No. If two angles are obtuse, which means more than \(90^{\circ}\), then they cannot be supplementary because their sum will always be more than \(180^{\circ}\).

(iii). Right?

Solution:-

Yes. If two angles are right, which means both measure \(90^{\circ}\), then they can form a supplementary pair.

∴ \(90^{\circ}\) + \(90^{\circ}\)= \(180^{\circ}\)

8. An angle is greater than \(45^{\circ}\). Is its complementary angle greater than \(45^{\circ}\) or equal to \(45^{\circ}\) or less than \(45^{\circ}\)?

Solution:-

Let us assume the complementary angles be \(p\) and \(q\),

We know that the sum of measures of complementary angle pair is \(90^{\circ}\).

\(= p + q = 90^{\circ}\)

It is given in the question that \(p > 45^{\circ}\)

Adding \(q\) on both sides,

\(= p + q > 45^{\circ} + q\)

\(= 90^{\circ} > 45^{\circ} + q\)

\(= 90^{\circ} – 45^{\circ} > q\)

\(= q < 45^{\circ}\)

Hence, its complementary angle is less than \(45^{\circ}\).9. 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 90o.

(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 180o.

(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.

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



(i) Obtuse vertically opposite angles
Solution:-

∠AOD and ∠BOC are obtuse vertically opposite angles.

(ii) Adjacent complementary angles

Solution:-

∠EOA and ∠AOB are adjacent complementary angles.

(iii) Equal supplementary angles

Solution:-

∠EOB and EOD are the equal supplementary angles.

(iv) Unequal supplementary angles

Solution:-

∠EOA and ∠EOC are the unequal supplementary angles.

(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.