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 90o.
The given angle is 20o
Let the measure of its complement be xo.
x + 20o = 90o
x = 90o – 20o
x = 70o
Hence, the complement of the given angle measures 70o.
(ii)
Solution:-
Two angles are said to be complementary if the sum of their measures is 90o.
The given angle is 63o
Let the measure of its complement be xo.
x + 63o = 90o
x = 90o – 63o
x = 27o
Hence, the complement of the given angle measures 27o.
(iii)
Solution:-
Two angles are said to be complementary if the sum of their measures is 90o.
The given angle is 57o
Let the measure of its complement be xo.
x + 57o = 90o
x = 90o – 57o
x = 33o
Hence, the complement of the given angle measures 33o.
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 180o.
The given angle is 105o
Let the measure of its supplement be xo.
x + 105o = 180o
x = 180o – 105o
x = 75o
Hence, the supplement of the given angle measures 75o.
(ii)
Solution:-
Two angles are said to be supplementary if the sum of their measures is 180o.
The given angle is 87o
Let the measure of its supplement be xo.
x + 87o = 180o
x = 180o – 87o
x = 93o
Hence, the supplement of the given angle measures 93o.
(iii)
Solution:-
Two angles are said to be supplementary if the sum of their measures is 180o.
The given angle is 154o
Let the measure of its supplement be xo.
x + 154o = 180o
x = 180o – 154o
x = 26o
Hence, the supplement of the given angle measures 93o.
3. Identify which of the following pairs of angles are complementary and which are supplementary.
(i) 65o, 115o
Solution:-
65o + 115o= 180o
If the sum of two angle measures is 180o, then the two angles are said to be supplementary.
∴ These angles are supplementary angles.
(ii) 63o, 27o
Solution:-
63o + 27o= 90o
If the sum of two angle measures is 90o, then the two angles are said to be complementary.
∴ These angles are complementary angles.
(iii) 112o, 68o
Solution:-
112o + 68o = 180o
If the sum of two angle measures is 180o, then the two angles are said to be supplementary.
∴ These angles are supplementary angles.
(iv) 130o, 50o
Solution:-
130o + 50o = 180o
If the sum of two angle measures is 180o, then the two angles are said to be supplementary.
∴ These angles are supplementary angles.
(v) 45o, 45o
Solution:-
45o + 45o = 90o
If the sum of two angle measures is 90o, then the two angles are said to be complementary.
∴ These angles are complementary angles.
(vi) 80o, 10o
Solution:-
80o + 10o = 90o
If the sum of two angle measures is 90o, 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 xo.
We know that the sum of measures of complementary angle pair is 90o.
Then,
= x + x = 90o
2x = 90o
\(x = {90^{\circ}\over 2}\)x = 45o
Hence, the required angle measure is 45o.
5. Find the angles which are equal to their supplement.
Solution:-
Let the measure of the required angle be xo.
We know that the sum of measures of supplementary angle pair is 180o.
x + x = 180o
2x = 180o
\(x = {180^{\circ}\over 2}\)
x = 90o
Hence, the required angle measure is 90o.
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, which means less than 90o, then they cannot be supplementary because their sum will always be less than 90o.
(ii). Obtuse?
Solution:-
No. If two angles are obtuse, which means more than 90o, then they cannot be supplementary because their sum will always be more than 180o.
(iii). Right?
Solution:-
Yes. If two angles are right, which means both measure 90o, then they can form a supplementary pair.
∴ 90o + 90o = 180
8. An angle is greater than 45o. Is its complementary angle greater than 45o or equal to 45o or less than 45o?
Solution:-
Let us assume the complementary angles be p and q,
We know that the sum of measures of complementary angle pair is 90o.
Then,
= p + q = 90o
It is given in the question that p > 45o
Adding q on both sides,
= p + q > 45o + q
= 90o > 45o + q
= 90o – 45o > q
= q < 45o
Hence, its complementary angle is less than 45o.
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 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.
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