NCERT Solutions Class 9 Chapter 6 Ex 6.1
The solutions to exercise 6.1 (from class 9th's maths NCERT book) are given below. The exercise has 6 questions which have been explained thoroughly with proper reasoning.
Given: ∠AOC + ∠EOB = 700 … (1)
Solution:
∠AOB is a straight angle and it is made up of 3 angles
∠AOB = 1800, or
∠AOC + ∠COE + ∠EOB = 1800
700 + ∠COE = 1800
∠COE = 1100
Reflex of ∠COE = 3600- 1100 => 2500
∠BOD = ∠AOC = 400 (Vertically opposite angles)
Using (1)
∠AOC + ∠EOB = 700
400 + ∠EOB = 700
∠EOB = 300
Let a and b be 2x and 3x
XOY is a straight line therefore ∠XOY = 1800
∠XOY can be written as sum of ∠XOM, ∠MOP and ∠POY, or ∠XON and ∠NOY
Considering the upper portion
∠XOY = 1800
∠XOM + ∠MOP + ∠POY = 1800
b+ a+ 90 = 1800
b +a = 900
2x +3x = 900
5x= 900
x= 180
Therefore a= 2x = 2(18) => 360 and b= 3x = 3(18) => 540
b= ∠NOY = 540 (Vertically opposite angles)
∠XON + ∠NOY = 1800 (linear pair)
c + 540 = 1800
c = 1260
Given: ∠ PQR = ∠ PRQ
∠ PQR + ∠ PQS = 1800 (Linear pair) … (1)
∠ PRT + ∠ PRQ = 1800 (Linear pair) … (2)
Equating (1) and (2)
∠ PQR + ∠ PQS = ∠ PRT + ∠ PRQ
∠ PQS = ∠ PRT + (∠ PRQ - ∠ PQR)
∠ PQS = ∠ PRT + 0 {∠ PQR = ∠ PRQ}
∠ PQS = ∠ PRT
Hence proved
x + y+ w+ z = 3600 (complete angle)
(x+ y) and (w+ z) can only be equal if AOB is a line
Only in that case x+ y = w+ z = 1800
If AOB is not a line, the measure of (x+ y) will be either greater or less than (w+ z)
Starting with ∠ ROS
∠ PQR = ∠ POS + ∠ ROS
900 = ∠ POS + ∠ ROS
∠ QOS = ∠ QOR +∠ ROS
∠ QOS = 900 +∠ ROS
∠ QOS - ∠ ROS = 900
Since I need ∠ POS, ∠ ROS and ∠ QOS, I am equating the above 2 equation instead of using substitution
∠ QOS - ∠ ROS = ∠ POS + ∠ ROS
∠ QOS - ∠ POS = ∠ ROS + ∠ ROS
2∠ ROS = ∠ QOS - ∠ POS
∠ ROS = 12(∠ QOS - ∠ POS)
Hence proved
∠ XYZ +∠ZYP = 1800 (Linear pair)
640 + +∠ZYP = 1800
∠ZYP = 1800 - 640
∠ZYP = 1160
YQ bisects ∠ZYP, therefore ∠ZYQ = ∠QYP = 12 x 1160= 580
∠XYQ = ∠ XYZ + ∠ZYQ => 640 + 580 => 1220
Reflex ∠QYP = 3600 - 580 => 3020
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