Mathematics Solutions Solutions for Class 7 Math Chapter 4 - Angles And Pairs Of Angles

Answer:

Answer:

Two angles which have a common vertex, a common arm and separate interiors are said to be adjacent angles.
The pairs of adjacent angles are given below:
∠ANB and ∠BNC,
∠BNC and ∠ANC,
∠ANC and ∠ANB,
∠PQR and ∠PQT

Answer:

Two angles which have a common vertex, a common arm and separate interiors are said to be adjacent angles
(i)
In ∠PMQ and ∠RMQ, M is the common vertex and MQ is the common arm.
Therefore, ∠PMQ and ∠RMQ are adjacent angles.     
(ii) 
The angles ∠RMQ and ∠SMR have a common vertex M, but don't have common arm.
Therefore, ∠RMQ and ∠SMR are not adjacent angles.     
(iii)
The angles ∠RMS and ∠RMT have a common vertex M, but don't have common arm.
Therefore, ∠RMS and ∠RMT are not adjacent angles.   
(iv)
In ∠SMT and ∠RMS, M is the common vertex and SM is the common arm.
Therefore, ∠SMT and ∠RMS are adjacent angles.

Answer:

(i)
Let the measure of the complementary angle be a.
40 + a = 90
 ∴ a = 50°
Hence, the measure of the complement of an angle of measure 40° is 50°
(ii)
Let the measure of the complementary angle be a.
63 + a = 90
 ∴ a = 27°
Hence, the measure of the complement of an angle of measure 63° is 27°
(iii)
Let the measure of the complementary angle be a.
45 + a = 90
 ∴ a = 45°
Hence, the measure of the complement of an angle of measure 45° is 45°
(iv)
Let the measure of the complementary angle be a.
55 + a = 90
 ∴ a = 35°
Hence, the measure of the complement of an angle of measure 55° is 35°
(v)
Let the measure of the complementary angle be a.
20 + a = 90
 ∴ a = 70°
Hence, the measure of the complement of an angle of measure 20° is 70°
(vi)
Let the measure of the complementary angle be a.
90 + a = 90
 ∴ a = 0°
Hence, the measure of the complement of an angle of measure 00° is 0°
(vii)
Let the measure of the complementary angle be a.
x + a = 90
 ∴ a = (90 − x)°
Hence, the measure of the complement of an angle of measure x° is (90 − x)°

Answer:

Sum of two complementary angles is 90^°
∴ (y $-$ 20)^° + (y + 30)^° = 90^°
⇒ y $-$ 20 + y + 30 = 90
⇒ 2y + 10 = 90
⇒ 2y = 80
⇒ y = 40
Hence, the measure of the two angles are 20^° and 70^°.
 

Answer:

(i)
Let the measure of the supplementary angle be a.
15 + a = 180
 ∴ a = 165°
Hence, the measure of the supplement of an angle of measure 15° is 165°.
(ii)
Let the measure of the supplementary angle be a.
85 + a = 180
 ∴ a = 95°
Hence, the measure of the supplement of an angle of measure 85° is 95°.
(iii)
Let the measure of the supplementary angle be a.
120 + a = 180
 ∴ a = 60°
Hence, the measure of the supplement of an angle of measure 120° is 60°.
(iv)
Let the measure of the supplementary angle be a.
37 + a = 180
 ∴ a = 143°
Hence, the measure of the supplement of an angle of measure 37° is 143°.
(v)
Let the measure of the supplementary angle be a.
108 + a = 180
 ∴ a = 72°
Hence, the measure of the supplement of an angle of measure 108° is 72°.
​(vi)
Let the measure of the supplementary angle be a.
0 + a = 180
 ∴ a = 180°
Hence, the measure of the supplement of an angle of measure 0° is 180°.
(vii)
Let the measure of the supplementary angle be x.
a + x = 180
 ∴ x = (180 − a)°
Hence, the measure of the supplement of an angle of measure a° is (180 − a)°.

Answer:

If the sum of the measures of two angles is 90° they are known as complementary angles. 
Hence,the pairs of complementary angles are ∠B and ∠N, ∠D and ∠F, ∠Y and ∠E.
If the sum of the measures of two angles is 180° they are known as supplementary angles. 
Hence, the pairs of supplementary angles are ∠B and ∠G, ∠N and ∠J.

Answer:

In ∆XYZ,
∠X + ∠Y + ∠Z = 180^°    (Angle Sum property of triangle
⇒ ∠X + 90^° + ∠Z = 180^°
⇒ ∠X + ∠Z = 90^°
Since, the sum of the measure of the two angles is 90^°.
Hence, ∠X and ∠Z are complementary angles.

Answer:

Let the measure of the first angle a.
Then, the measure of the other angle a + 40°
Now, a + a + 40 = 90
⇒ 2a = 50
⇒ a = 25°
Hence, the measure of the two angles are 25° and 65°.

Answer:

If the sum of the measures of two angles is 180° they are known as supplementary angles. 
The measure of all the angles of a rectangle is 90°.
Hence, the pairs of supplementary angles are ∠P and ∠M, ∠T and ∠N, ∠P and ∠T, ∠M and ∠N, ∠P and ∠N, ∠M and ∠T.

Answer:

Let the measure of the complementary angle be a.
70 + a = 90
 ∴ a = 20°
Let the measure of the supplementary angle of 20° be x.
20 + x = 180
 ∴ x = 160°
Hence, the measure of the supplement of the complement of ∠A is 160°.
 

Answer:

Let the measure of the supplementary angle of ∠B be a.
(x + 20)^° + a = 180
 ∴ a = (160 − x)°
Hence, the measure of ∠A is (160 − x​)°.

Answer:

Two rays which have a common origin and form a straight line are said to be opposite rays. 
Hence, the pairs of opposite rays are ray PL & ray  PM and ray PN & ray PT.

Answer:

Ray PM and PT are not opposite rays because they do not form a straight line.

Answer:

(i)  


(ii)   

If the sum of the measures of two angles is 180° they are known as supplementary angles.
The sum of the measures of the angles in a linear pair is 180°.
Therefore, angles in a linear pair are always supplementary.

(iii)   

(iv)  

(v)   

Answer:

In the given figure, 
∠DPA + ∠APB = 180^∘         (Linear Pair angles)
⇒ 47^∘ + ∠APB = 180^∘
⇒ ∠APB = 133^∘
Now, 
∠APD = ∠BPC = 47^∘          (Vertically opposite angles)
∠APB = ∠DPC = 133^∘          (Vertically opposite angles)
Hence, the measures of ∠APB, ∠BPC, ∠CPD are 133^∘, 47^∘ and 133^∘ respectively.

Answer:

In the given figure, 
∠RMP + ∠PMS = 180^∘         (Linear Pair angles)
⇒ x^∘ + ∠PMS = 180^∘
⇒ ∠PMS = (180 − x)^∘
Now, 
∠PMR = ∠SMQ = x^∘                       (Vertically opposite angles)
∠PMS = ∠RMQ = (180 − x)^∘          (Vertically opposite angles)
Hence, the measures of ∠PMS, ∠SMQ and ∠QMR are (180 − x)^∘, x^∘ and (180 − x)^∘ respectively.

Answer:

∠A + ∠B = ∠ACD   (Exterior angle property)
⇒ 2∠A = 140^∘         (∵∠A = ∠B)        
⇒ ∠A = 70^∘
Hence, the measures of ∠A and ∠B are 70^∘ and 70^∘ respectively.

Answer:

In the given figure, 
∠BOC = ∠FOE = 4y           (Vertically opposite angles)
∠EOD = ∠AOB = 8y           (Vertically opposite angles)
∠AOF = ∠COD = 6y           (Vertically opposite angles)
Now, ∠AOB + ∠BOC + ∠COD = 180^∘         (Linear Pair angles)
⇒ 8y + 4y + 6y = 180^∘
⇒ 18y = 180^∘
⇒ y = 10^∘
Therefore, 
∠BOC = 4y
= 40^∘
∠EOD = 8y
= 80^∘
​∠AOF = 6y
= 60^∘
Hence, the measures of ∠BOC, ∠EOD, ∠AOF are 40^∘, 80^∘ and 80^∘ respectively.

Answer:

Given:
∠ACB = (3x $-$ 17)^∘
∠ACD = (8x + 10)^∘
Now, ∠ACB + ∠ACD = 180^∘         (Linear Pair angles)
⇒ 3x − 17 + 8x + 10 = 180
⇒ 11x = 187
⇒ x = 17
Therefore,
∠ACB = (3x $-$ 17)^∘
= (51 $-$ 17)^∘
= 34^∘
​∠ACD = (8x + 10)^∘
= (136 + 10)^∘
= 146^∘
Now, ∠A + ∠B = ∠ACD   (Exterior angle property)
⇒ 2∠A = 146^∘         (∵∠A = ∠B)        
⇒ ∠A = 73^∘
Hence, the measures of ∠ACB, ∠ACD, ∠A and ∠B are 146^∘, 34^∘, 73^∘ and 73^∘ respectively.