**Lines and Angles**

**Exercise 5.1**

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

i) 20^{0}

ii). 63^{0}

iii). 57^{0}

Solution:

i) 20^{0}

let the complementary of 20^{0} be *x*.

*x* + 20 = 90

*x* = 90 – 20

*x* = 70

ii). 63^{0}

let the complementary of 63^{0} be *x*.

*x* + 63 = 90

*x* = 90 – 63

*x* = 27

iii). 57^{0}

let the complementary of 57^{0} be *x*.

*x* + 57 = 90

*x* = 90 – 57

*x* = 33

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

i) 105^{0}

ii). 87^{0}

iii). 154^{0}

Solution:

i) 105^{0}

let the supplementary of 105^{0} be *x*.

*x* + 105 = 180

*x* = 180 – 105

*x* = 75

ii). 87^{0}

let the supplementary of 87^{0} be *x*.

*x* + 87 = 180

*x* = 180 – 87

*x* = 93

iii). 154^{0}

let the supplementary of 154^{0} be *x*.

*x* + 154 = 180

*x* = 180 – 154

*x* = 26

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

(i) 65º, 115º (ii) 63º, 27º (iii) 112º, 68º

(iv) 130º, 50º (v) 45º, 45º (vi) 80º, 10º

Solution:

(i) 65º, 115º

The Sum of the two angles is 65 + 115 = 180

The given pair of angles are supplementary.

(ii) 63º, 27º

The Sum of the two angles is 63 + 27 = 90

The given pair of angles are complementary.

(iii) 112º, 68º

The Sum of the two angles is 112 + 68 = 180

The given pair of angles are supplementary.

(iv) 130º, 50º

The Sum of the two angles is 130 + 50 = 180

The given pair of angles are supplementary.

(v) 45º, 45º

The Sum of the two angles is 45 + 45 = 90

The given pair of angles are complementary.

(vi) 80º, 10º

The Sum of the two angles is 80 + 10 = 90

The given pair of angles are complementary.

4). Find the angle which is equal to its complement.

The angle which is equal to its complement is 45.

5). Find the angle which is equal to its supplement.

The angle which is equal to its supplement is 90.

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 the angles still remain supplementary?

In supplementary angle, the sum of angles is always equal to 180.

∠1 and ∠2 are supplementary angles.

If ∠1 is decreased, then ∠2 will increase so that the sum is always equal to 180.

7). Can two angles be supplementary if both of them are:

(i) acute? (ii) obtuse? (iii) right?

Answer

(i) acute?

Two angles cannot be supplementary if both of them are acute.

(ii) obtuse?

Two angles cannot be supplementary if both of them are obtuse.

(iii) right?

Two angles are supplementary if both of them are right angles.

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

In complementary angles the sum of two angles is always equal to 90.

Therefore, if one angle is greater than 45^{0} then the complement is less than 45^{0}.

9). In the adjoining figure:

(i) Is ∠1 adjacent to ∠2?

Yes.

(ii) Is ∠AOC adjacent to ∠AOE?

No.

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

Yes.

(iv) Are ∠BOD and ∠DOA supplementary?

Yes.

(v) Is ∠1 vertically opposite to ∠4?

Yes.

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

∠COB (∠2 + ∠3)

10). Indicate which pairs of angles are:

(i) Vertically opposite angles. (ii) Linear pairs.

Answer:

(i) Vertically opposite angles.

∠5 and ∠2 + ∠3

∠1 and ∠4

(ii) Linear pairs.

(∠5 and ∠4), (∠5 and ∠1), (∠1, ∠2 and ∠3), and (∠2, ∠3 and ∠4) are linear pairs.

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

∠1 is not adjacent to ∠2 because they don’t have a common vertex.

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

i). 55^{0} and ∠x are vertically opposite angles

∠x = 55^{0}

55^{0} and ∠y are forming linear pair

55^{0} + ∠y = 180

∠y = 180 – 55^{0}

∠y = 125^{0}

∠y and ∠z are vertically opposite angles

∠y = 125^{0}

∠z = 125^{0}

Ans: ∠x = 55^{0}

∠y = 125^{0}

∠z = 125^{0}

ii). 40^{0} and ∠y are forming linear pair

40^{0} + ∠y = 180

∠y = 180 – 40

∠y = 140^{0}

∠z and ∠y are forming linear pair

∠z + ∠y = 180

∠z + 140 = 180

∠z = 180 – 140

∠z = 40^{0}

40^{0}, ∠x and ∠y are forming linear pair

40 + 40 + ∠y = 180

∠x + 80 = 180

∠x = 180 – 80

∠x = 100^{0}

Ans: ∠x = 100^{0}

∠y = 140^{0}

∠z = 40^{0}

13). Fill in the blanks:

(i) If two angles are complementary, then the sum of their measures is __90__.

(ii) If two angles are supplementary, then the sum of their measures is __180__.

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

(iv) 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 __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 __obtuse__.

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

(i) Obtuse vertically opposite angles

(ii) Adjacent complementary angles

(iii) Equal supplementary angles

(iv) Unequal supplementary angles

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

Answer:

(i) Obtuse vertically opposite angles

∠AOD and ∠BOC

(ii) Adjacent complementary angles

∠AOE and ∠AOB

(iii) Equal supplementary angles

∠AOE and ∠EOD

(iv) Unequal supplementary angles

∠BOC and ∠COD

∠BOA and ∠AOD

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

∠AOE and ∠AOB

∠AOE and ∠EOC

∠DOE and ∠DOC

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