Chapter 8 Lines and Angles RD Sharma Solutions Exercise 8.3 Class 9 Maths

 Chapter Name RD Sharma Chapter 8 Lines and Angles Exercise 8.3 Book Name RD Sharma Mathematics for Class 10 Other Exercises Exercise 8.1Exercise 8.2Exercise 8.4 Related Study NCERT Solutions for Class 10 Maths

Exercise 8.3 Solutions

1. In the below fig, lines l1 and l2 intersect at O, forming angles as shown in the figure. If x = 45, Find the values of x, y, z and u.

Solution

Given that
x = 45° , y = ? , z = ?, u = ?
Vertically opposite sides are equal
∴ z = x = 45°
z and u angles are linear pair of angles
∴ z + u = 180°
z = 180° - 4
⇒ u = 180° - x
⇒ u = 180° - 45° [∵ x = 45°]
⇒ u = 135°
x and y angles are linear pair of angles
∴ x + y = 180°
y = 180° - x
⇒ y = 180° - 45°
⇒ y = 135°
∴ x = 45°, y = 135°, z = 135° and u = 45°

2. In the below fig, three coplanar lines intersect at a point O, forming angles as shown in the figure. Find the values of x, y, z and u.

Solution
Vertically opposite angles are equal
So ∠BOD = z = 90°
∠ DOF = y = 50°
Now, x + y + z = 180°  [Linear pair]
⇒ x + y + z = 180°
⇒ 90° + 50° + x = 180°
⇒ x = 180° - 140°
⇒ x = 40°

3. In the given fig, find the values of x, y and z.
Solution
From the given figure
∠y = 25° [∵ vertically opposite angles are equal]
Now
∠x + ∠y = 180° [Linear pair of angles are x and y]
⇒ ∠x = 180° - 25°
⇒ ∠x = 155°
Also
∠z = ∠x  155° [Vertically opposite angle]
∠y = 25°
∠x = ∠z = 155°

4. In the below fig, find the values of x.
Solution
Vertically opposite angles are equal
∠AOE = ∠BOF = 5x
Linear pair
∠COA + ∠AOE + ∠EOD = 180°
⇒ 3x + 5x + 2x = 180°
⇒ 10x = 180°
⇒ x = 18°

5. Prove that the bisectors of a pair of vertically opposite angles are in the same straight line.
Solution
Given,
Lines AOB and COD intersect at point O such that
∠AOC = ∠BOD
Also OE is the bisector ∠ADC and OF is the bisector ∠BOD
To prove : EOF is a straight line vertically opposite angles is equal
∠AOD = ∠BOC = 5x  ...(1)
Also ∠AOC + ∠BOD
⇒ 2∠AOE  = 2∠DOF ...(2)
Sum of the angles around a point is 360°
⇒ 2∠AOD + 2∠AOE + 2∠DOF = 360
⇒ ∠AOD + ∠AOF + ∠DOF = 180
From this we conclude that EOF is a straight line.
Given that : - AB and CD intersect each other at O
OE bisects ∠COB
To prove: ∠AOF = ∠DOF
Proof: OE bisects ∠COB
∠COE = ∠EOB = x
Vertically opposite angles are equal
∠BOE = ∠AOF  = x  ...(1)
∠COE = ∠DOF = x  ...(2)
From (1) and (2)
∠AOF = ∠DOF = x

6. If one of the four angles formed by two intersecting lines is a right angle, then show that each of the four angles is  a right angle.
Solution
Given, AB and CD are two lines intersecting at O such that
∠ BOC = 90°
∠AOC = 90° , ∠AOD = 90° and ∠BOD = 90°
Proof:
Given that ∠BOC = 90°
Vertically opposite angles are equal

7. In the below fig, rays AB and CD intersect at O.
(i) Determine y when x = 60°
(ii) Determine x when y = 40°
Solution
(i) Given x = 60°
y = ?
∠AOC, ∠BOC are linear pair of angles
∠AOC + ∠BOC = 180°
⇒ 2x + y = 180°
⇒ 2×60° + y = 180°  [∵ x = 60°]
⇒ y = 180° - 120°
⇒ y = 60°
(ii) Given y = 40°, x = ?
∠AOC and ∠BOC are linear pair of angles
∠AOC + ∠BOC = 180°
2x + y = 180°
⇒ 2x + 40 = 180°
⇒ 2x = 140°
⇒ x = 140°/2
⇒ x = 70°

8.In the below fig, lines AB, CD and EF intersect at O. Find the measures of ∠AOC, ∠COF, ∠DOE and ∠BOF.
Solution
∠AOE and ∠EOB are linear pair of angles

9. AB, CD and EF are three concurrent lines passing through the point O such that OF bisects ∠BOD. If ∠BOF = 35 ,find ∠BOC and ∠AOD.
Solution

10. In below figure, lines AB and CD intersect at O. If ∠AOC + ∠BOE = 70° and ∠BOD = 40°, find ∠BOE and reflex ∠COE.

Solution
Given that
∠AOC + ∠BOE = 70° and ∠BOD = 40°
∠BOE = ?
Here, ∠BOD and ∠AOC are vertically opposite angles
∠BOD = ∠AOC = 40°
Given ∠AOC + ∠BOE = 70°
40° + ∠BOF = 70°
∠BOF = 70°- 40°
∠BOE = 30°
∠AOC and ∠BOC are linear pair of angles
∠AOC + ∠COF + ∠BOE  = 180°
∠COE = 180° - 30° - 40°
⇒ ∠COE = 110°
∴ Reflex ∠COE = 360° - 110° = 250°.

11. Which of the following statements are true (T) and which are false (F) ?
(i) Angles forming a linear pair are supplementary.
(ii) If two adjacent angles are equal, and then each angle measures 90° .
(iii) Angles forming a linear pair can both the acute angles.
(iv) If angles forming a linear pair are equal, then each of these angles is of measure 90° .
Solution
(i) True
(ii) False
(iii) False
(iv)  True

12. Fill in the blanks so as to make the following statements true:
(i) If one angle of a linear pair is acute, then its other angle will be __________
(ii) A ray stands on a line, then the sum of the two adjacent angles so formed is ________.
(iii) If the sum of two adjacent angles is 180° , then the ________ arms of the two angles are opposite rays.
Solution
(i) Obtuse angle
(ii) 180°
(iii) uncommon