Ch.4 Angles and Pairs of angles

04.Angles and Pairs of Angles

TextBook Page No 25:

PRACTICE SET 15

Question 1:

Observe the figure and complete the table for ∠AWB.

 

Points in the interior
Points in the exterior
Points on the arms of the angles

 

ANSWER:

Points in the interior	R, C, N, X
Points in the exterior	T, U, Q, V, Y
Points on the arms of the angles	A, W, G, B

Page No 25:

Question 2:

Name the pairs of adjacent angles in the figures below.

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

Page No 25:

Question 3:

Are the following pairs adjacent angles? If not, state the reason.

(i) ∠PMQ and ∠RMQ        (ii) ∠RMQ and ∠SMR

(iii) ∠RMS and ∠RMT      (iv) ∠SMT and ∠RMS

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.

Page No 26:

PRACTICE SET 16

Question 1:

The measures of some angles are given below. Write the measures of their complementaryangles.

(i) 40^°   (ii) 63^° (iii) 45^°(iv) 55^° (v) 20^° (vi) 90^°  (vii) x^°

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)°

Page No 26:

Question 2:

(y − 20)^° and (y + 30)^° are the measures of complementary angles. Find the measure of each angle.

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

substitute the value of Y

y - 20

= 40 - 20

= 20

------------

y + 30

40 + 30

= 70

Hence, the measure of the two angles are 20^° and 70^°.
 

Page No 27:

PRACTICE SET NO.17

Question 1:

Write the measures of the supplements of the angles given below.

(i) 15^°  (ii) 85^° (iii) 120^° (iv) 37^° (v) 108^°  (vi) 0^° (vii) a^°

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)°.

Page No 27:

Question 2:

The measures of some angles are given below. Use them to make pairs of  complementary and supplementary angles.

m∠B = 60^° m∠N = 30^° m∠Y = 90^°  m∠J = 150^°
m∠D = 75^° m∠E = 0^° m∠F = 15^° m∠G = 120^°

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.

Page No 27:

Question 3:

In ∆XYZ, m∠Y = 90^°. What kind of a pair do ∠X and ∠Z make?

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.

Page No 27:

Question 4:

The difference between the measures of the two angles of a complementary pair is  40^°. Find the measures of the two 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°.

Page No 27:

Question 5:

□$\square$ PTNM is a rectangle. Write the names of the pairs of supplementary angles.

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.

Page No 27:

Question 6:

If m∠A = 70^°, what is the measure of the supplement of the complement of ∠A?

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°.
 

Page No 27:

Question 7:

If ∠A and ∠B are supplementary angles and m∠B = (x + 20)^°, then what would be m∠A?

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​)°.

Page No 28:

Practice SetNo.18

Question 1:

Name the pairs of opposite rays in the figure alongside.

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.

Page No 28:

Question 2:

Are the ray PM and PT opposite rays?  Give reasons for your answer.

ANSWER:

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

Page No 29:

PRACTICE SET NO.19

Question 1:

Draw the pairs of angles as described below.If that is not possible, say why.

(i)  Complementary angles that are not adjacent.

(ii)   Angles in a linear pair which are not supplementary.

(iii)   Complementary angles that do not form a linear pair.

(iv)  Adjacent angles which are not in a linear pair.

(v)   Angles which are neither complementary nor adjacent.

(vi)   Angles in a linear pair which are complementary.

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)   

(vi)   
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.

Page No 30:

PRACTICE SET NO.20

Question 1:

Lines AC and BD intersect at point P. m∠APD = 47^° .Find the measures of ∠APB, ∠BPC, ∠CPD.

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.

Page No 30:

Question 2:

Lines PQ and RS intersect at point M. m∠PMR = x^° What are the measures of ∠PMS, ∠SMQ and ∠QMR?


 

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.

Page No 33:

PRACTICE SET NO.21

Question 1:

∠ACD is an exterior angle of Δ$∆$ABC.The measures of ∠A and ∠B are equal. If m∠ACD = 140^°, find the measures of the angles ∠A and ∠B.

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.

Page No 33:

Question 2:

Using the measures of the angles given in the figure alongside, find the measures of the remaining three angles.

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.

Page No 33:

Question 3:

In the isosceles triangle ABC, ∠A and ∠B are equal. ∠ACD is an exterior angle of ∆ABC. The measures of ∠ACB and ∠ACD are
(3x−$-$17)^° and (8x + 10)^° respectively. Find the measures of ∠ACB and ∠ACD. Also find the measures of ∠A and ∠B.

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 −$17)^\circ$= (51 −17)$^<br>$= 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.