Chapter 5 Lines and Angles Exercise 5.1

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

Solution:
(i) Complement of 20° = 90° – 20° = 70°
(ii) Complement of 63° = 90° – 63° = 27°
(iii) Complement of 57° = 90° – 57° = 33°

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

Solution:
(i) Supplement of 105° = 180° – 105° = 75°
(ii) Supplement of 87° = 180° – 87° = 93°
(iii) Supplement of 154° = 180° – 154° = 26°

Question 3.
Identify which of the following pairs of angles are complementary and which are supplementary?
(i) 65°, 115°
(ii) 63°, 27°
(iii) 112°, 68°
(iv) 130°, 50°
(v) 45°, 45°
(vi) 80°, 10°
Solution:
(i) 65° (+) 115° = 180°
They are supplementary angles.
(ii) 63° (+) 27° = 90°
They are complementary angles.
(iii) 112° (+) 68° = 180°
They are supplementary angles.
(iv) 130° (+) 50° = 180°
They are supplementary angles.
(v) 45° (+) 45° = 90°
They are complementary angles.
(vi) 80° (+) 10° = 90°
They are complementary angles.

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.

Solution:
∠1 + ∠2 = 180° (given)
If ∠1 is decreased by some degrees, then ∠2 will also be increased by the same degree so that the two angles still remain supplementary.

Question 7.
Can two angles be supplementary if both of them are:
(i) acute?
(ii) obtuse?
(iii) right?
(ii) Since, acute angle < 90°
∴ Acute angle + acute angle < 90° + 90° < 180° Thus, the two acute angles cannot be supplementary angles. (ii) Since, obtuse angle > 90°
∴ Obtuse angle + obtuse angle > 90° + 90° > 180°
Thus, the two obtuse angles cannot be supplementary angles.
(iii) Since, right angle = 90°
∴ right angle + right angle = 90° + 90° = 180°
Thus, two right angles are supplementary angles.

Question 8.
An angle is greater than 45°. Is its complementary angle greater than 45° or equal to 45° or less than 45 °?
Solution:
Given angle is greater than 45°
Let the given angle be x°.
∴ x > 45
Complement of x° = 90° – x° < 45° [ ∵ x > 45°]
Thus the required angle is less than 45°.

Question 9.
In the following figure:
(i) Is ∠1 adjacent to ∠2?
(iii) Do ∠COE and ∠EOD form a linear pair?
(iv) Are ∠BOD and ∠DOA supplementary?
(v) Is ∠1 vertically opposite angle to ∠4?
(vi) What is the vertically opposite angle of ∠5?

Solution:
(i) Yes, ∠1 and ∠2 are adjacent angles.
(ii) No, ∠AOC is not adjacent to ∠AOE. [ ∵  OC and OE do not lie on either side of common arm OA] .
(iii) Yes, ∠COE and ∠EOD form a linear pair of angles.
(iv) Yes, ∠BOD and ∠DOA are supplementary. [∵ ∠BOD + ∠DOA = 180°]
(v) Yes, ∠1 is vertically opposite to ∠4.
(vi) Vertically opposite angle of ∠5 is ∠2 + ∠3 i.e. ∠BOC.

Question 10.
Indicate which pairs of angles are:
(i) Vertically opposite angles
(ii) Linear pairs
Solution:
(i) Vertically opposite angles are ∠1 and ∠4, ∠5 and (∠2 + ∠3)

(ii) Linear pairs are
∠1 and ∠5, ∠5 and ∠4

Question 11.
In the following figure, is ∠1 adjacent to ∠2? Give reasons.
Solution:
No, ∠1 and∠2 are not adjacent angles.

Reasons:
(i) ∠1 + ∠2 ≠ 180°
(ii) They have no common vertex.

Question 12.
Find the values of the angles x, y and z in each of the following: Solution:
From Fig. 1. we have
∠x = ∠55° (Vertically opposite angles)
∠x + ∠y = 180° (Adjacent angles)
55° + ∠y = 180° (Linear pair angles)
∴ ∠y = 180° – 55° = 125°
∠y = ∠z (Vertically opposite angles)
125° = ∠z
Hence, ∠x = 55°, ∠y = 125° and ∠z = 125°

(ii) 25° + x + 40° = 180° (Sum of adjacent angles on straight line)
65° + x = 180°
∴ x = 180° – 65° = 115°
40° + y = 180° (Linear pairs)
∴ y = 180° – 40° = 140°
y + z = 180° (Linear pairs)
140° + z = 180°
∴ z = 180° – 140° = 40°
Hence, x – 115°, y = 140° and z – 40°

Question 13.
Fill in the blanks:
(i) If two angles are complementary, then the sum of their measures is ______ .
(ii) If two angles are supplementary, then the sum of their measures is ______ .
(iii) Two angles forming a linear pair are ______ .
(iv) If two adjacent angles are supplementary, they form a ______ .
(v) If two lines intersect at a point, then the vertically opposite angles are always ______ .
(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:
(i) 90°
(ii) 180°
(iii) Supplementary
(iv) Linear pair
(v) Equal
(vi) Obtuse angle

Question 14.
In the given figure, name the following pairs of angles.

(i) Obtuse vertically opposite angles.