# Exercise 5.1

QUESTION 1

Find the complement of each of the following angles: The sum of the measures of complementary angles is 90°.

(i) 20°

Complement = 90° – 20°

= 70°

(ii) 63°

Complement = 90° – 63°

= 27°

(iii) 57°

Complement = 90° – 57°

= 33°

QUESTION 2

Find the supplement of each of the following angles: The sum of the measures of supplementary angles is 180°

(i) 105°

Supplement = 180° – 105°

= 75°

(ii) 87°

Supplement = 180° – 87°

= 93°

(iii) 154°

Supplement = 180° — 154°

= 26°

QUESTION 3

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

NOTE : The sum of the measures of complementary angles is 90° and that of supplementary
angles is 180°.

(i) 65°,115°

Sol :

Sum of the measures of these angles = 65° +115° = 180°

∴These angles are supplementary angles.

(ii) 63°, 27°

Sol :

Sum of the measures of these angles = 63° + 27° = 90°

∴These angles are complementary angles.

(iii) 112°, 68°

Sol :

Sum of the measures of these angles = 112° + 68° = 180°

∴These angles are supplementary angles.

(iv) 130°, 50°

Sol :

Sum of the measures of these angles = 130° + 50° = 180°

∴These angles are supplementary angles.

(v) 45°, 45°

Sol :

Sum of the measures of these angles = 45° + 45° = 90°

∴These angles are complementary angles.

(vi) 80°, 10°

Sol :

Sum of the measures of these angles = 80° + 10° = 90°

∴These angles are complementary angles.

QUESTION 4

Find the angles which is equal to its complement.

Sol :

Let the angle be x

Complement of this angle is also x

The sum of the measures of a complementary angle pair is 90°

x + x = 90°

2x= 90°

$\dfrac{90°}{2}$

= 45°

QUESTION 5

Find the angles which is equal to its supplement.

Sol :

Let the angle be x

Supplement of this angle is also x

The sum of the measures of a supplementary angle pair is 180°

x + x = 180°

2x = 180°

x = 90°

QUESTION 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. Sol :

∠1 and ∠2 are supplementary angles.

If ∠1 is reduced, then ∠2 should be increased by the same measure so that this angle pair remains supplementary

QUESTION 7

Can two angles be supplementary if both of them are:

(i) Acute ? (ii) Obtuse ? (iii) Right ?

Sol :

(i) No. Acute angle is always lesser than 90°. It can be observed that two angles, even of 89°, cannot add up to 180°. Therefore, two acute angles cannot be in a supplementary angle pair.

(ii) No. Obtuse angle is always greater than 90°. It can be observed that two angles, even of 91°, will always add up to more than 180°. Therefore, two obtuse angles

(iii) Yes. Right angles are of 90° and 90° + 90° = 180° . Therefore, two right angles form a supplementary angle pair together.

QUESTION 8

An angle is greater than 45°. ls its complementary angle greater than 45° or equal to 45° or less than 45° ?

Sol :

Let A and B are two angles making a complementary angle pair and A is greater than 45°

A + B = 90°

B = 90° – A

Therefore, 8 will be lesser than 45°

QUESTION 9

(i) Is ∠1 adjacent to ∠2 ?

(ii) Is ∠A0C adjacent to ∠AOE ?

(iii) Do ∠COE and ∠E0D form a linear pair ?

(iv) Are ∠BOD and ∠DOA supplementary ?

(v) Is ∠1 vertically opposite to ∠4 ?

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

(i) Yes, Since they have a common vertex O and also a common arm OC. Also, their non-common arms, OA and OE , are on either side of the common arm.

(ii) No, They have a common vertex O and also a common arm OA. However, their non-common arms, OC and OE, are on the same side of the common arm. Therefore, these are not adjacent to each other.

(iii) Yes, Since they have a common vertex O and a common arm OE. Also, their non-common arms, OC and OD, are opposite rays.

(iv) Yes, Since ∠BOD and ∠DOA have a common vertex O and their non-common arms are opposite to each other.

(v) Yes, Since these are formed due to the intersection of two straight lines (AB and CD )

(vi) ∠COB is the vertically opposite angle of ∠5 as these are formed due to the intersection of two straight lines, AB and CD.

QUESTION 10

Indicate which pairs of angles are:

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

(i) ∠1 and ∠4, ∠5 and ∠2 +∠3 are vertically opposite angles as these are formed due to the intersection of two straight lines.

(ii) ∠1 and ∠5, ∠5 and ∠4 are linear pairs as these have a common vertex and also have non-common arms opposite to each other , also they have a common vertex .

QUESTION 11

In the following figure, is ∠1 adjacent to ∠2 ? Give reasons. Sol :

∠1 and ∠2 are not adjacent angles because their vertex is not common.

QUESTION 12

Find the value of the angles x , y, and z in each of the following:

(i) Sol :

Since ∠x and ∠55° are vertically opposite angles,

∠x = ∠55°

∠x + ∠y = 180° (Linear pair)

∠55° + ∠y = 180°   (∠x = ∠55°)

∠y = 180° – ∠55°

∠y = 125°

∠y = ∠z  (Vertically opposite angles)

∠z =125°

(ii) Sol :

∠z = 40° (Vertically opposite angles)

∠y+ ∠z= 180° (Linear pair)

∠y = 180° – 40°

∠y = 140°

40° + ∠x+ 25° = 180° (Angles on a straight line)

65° + ∠x = 180°

∠x = 180° – 65°

∠x = 115°

QUESTION 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 angle .

QUESTION 14

In the adjoining figure, name the following pairs of angles. (i) Obtuse vertically opposite angles

Sol :

∠AOD , ∠BOC

Sol :

∠EOA , ∠AOB

(iii) Equal supplementary angles

Sol :

∠EOB , ∠EOD

(iv) Unequal supplementary angles

Sol :

∠EOA , ∠EOC

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

Sol :

∠AOB and ∠AOE , ∠AOE and ∠EOD , ∠EOD and ∠COD

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