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• RD Sharma Class 9 Solutions

Class 9 RD Sharma Solutions – Chapter 8 Introduction to Lines and Angles- Exercise 8.3

Question 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°

Find: the value of Y, Z, and u

z = x = 45° [Vertically opposite angles are equal]

z + u = 180° [z and u are angles that are a linear pair]

z = 180° – u

u = 180° – x

u = 180° – 45°

u = 135°

x + y = 180° [x and y angles are a linear pair]

y = 180° – x

y =180° – 45°

y = 135°

Hence, x = 45°, y = 135°, z = 135° and u = 45°

Question 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:

From the given figure

∠SOD = z = 90°     [Vertically opposite angles are equal]

∠DOF = y = 50°

Now, x + y + z = 180°   [Linear pair of angles]

Now put the value of z and y

90° + 50° + x = 180°

x = 180° – 140°

x = 40°

So, x = 40°, y = 50°, z = 90°, u = 40°

Question 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]

x = 180° – 25°

x = 155°

Also,

z = x = 155°    [Vertically opposite angles]

y = 25°

z = 155°

So, x = 155°, y = 25°, z = 155°

Question 4. In the below fig. find the value of x?

Solution:

From the figure

AOE = BOF = 5x    [Vertically opposite angles are equal]

∠COA + ∠AOE + ∠EOD = 180°   [Linear pair angles]

3x + 5x + 2x = 180°

10x = 180°

x = 180°/10

x = 18°

So, the value of x = 18°

Question 5. Prove that bisectors of a pair of vertically opposite angles are in the same straight line.

Solution:

From the figure

Lines AB and CD intersect at point O, such that

∠AOC = ∠BOD    [vertically opposite angles are equal] …(1)

Also, OP is the bisector of AOC and OQ is the bisector of BOD

To Prove: POQ is a straight line.

∠AOP = ∠COP    [OP is the bisector of ∠AOC]…(2)

∠BOQ = ∠QOD   [OQ is the bisector of ∠BOD]…(3)

Now,

∠AOC + ∠BOD + ∠AOP + ∠COP + ∠BOQ + ∠QOD = 360°        [sum of all angles around a point is 360°]

∠BOQ + ∠QOD + ∠DOA + ∠AOP + ∠POC + ∠COB = 360°

2∠QOD + 2∠DOA + 2∠AOP = 360° (from eq(1), (2) and (3))

∠QOD + ∠DOA + ∠AOP = 180°

POQ = 180°

Hence proved

Question 6. If two straight lines intersect each other, prove that the ray opposite to the bisector of one of the angles thus formed bisects the vertically opposite angle.

Solution:

Let us considered AB and CD intersect at a point O

Now draw the bisector OP of AOC

OP = POC … (i)

Let extend OP to Q.

Show that, OQ bisects BOD

Let us considered that OQ bisects BOD,

Prove that POQ is a line.

As we know that,

AOC = DOB  …. (ii)  [vertically opposite angles.]

AOP = BOQ   [vertically opposite angles.]

Similarly, POC = DOQ

AOP + AOD + DOQ + POC + BOC + BOQ = 360°  [sum of all angles around a point is 360 degrees ]

2AOP + AOD + 2D0Q + BOC = 360°

2AOP + 2AOD + 2DOQ = 360°

2(AOP + AOD + DOQ) = 360°

AOP + AOD +DOQ = 360°/2

AOP + AOD + DOQ = I80°

Thus, POQ is a straight line.

Hence proved

Question 7. 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:

According to question

AB and CD intersecting at O, such that ∠BOC = 90°, ∠AOC = 90 °∠AOD = 90° and ∠BOD = 90°

Given:∠BOC = 90°

∠BOC = ∠AOD = 90°    [Vertically opposite angles are equal]

∠AOC + ∠BOC = 180°    [Angles in linear pair]

∠AOC + 90°  = 180°  [Angles in linear pair]

∠AOC = 90°

∠AOC = ∠BOD = 90°    [Vertically opposite angles]

Hence, ∠AOC = ∠BOC = ∠BOD = ∠AOD = 90°

Question 8. 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 that x = 60°

∠AOC + ∠BOC = 180°   [linear pair of angles]

⟹ 2x + y = 180°

⟹ 2(60°) + y = 180° [since x = 60°]

⟹ y = 60°

Hence, the value of y = 60° when x = 60°

(ii) Given y = 40°

∠AOC + ∠BOC = 180°   [linear pair of angles]

⟹ 2x + y = 180°

⟹ 2x + 40° = 180°  [since x = 40°]

⟹ 2x =180° – 140°

⟹ 2x = 140°

⟹ x = 70°

Hence, the value of x = 70° when x = 40°

Question 9. In the below fig. lines AB. CD and EF intersect at O. Find the measures of ∠AOC, ∠COF, ∠DOE, and ∠BOF.

Solution:

From the figure

∠AOE + ∠EOB = 180°   [linear pair of angles]

∠AOE + ∠DOE + ∠BOD = 180°  [linear pair of angles]

⟹ ∠DOE = 180° – 40° – 35° = 105°

∠DOE = ∠COF = 105°    [Vertically opposite angles are equal]

Now, ∠AOE + ∠AOF = 180°   [Angles in Linear pair]

∠AOE + ∠AOC + ∠COF = 180°

⟹ 40° + ∠AOC +105° = 180°

⟹ ∠AOC = 180° – 145°

⟹ ∠AOC = 35°

Also, ∠BOF = ∠AOE = 40° [Vertically opposite angles are equal]

Hence, the value of ∠AOC = 35°, ∠COF = 105°, ∠DOE = 105°, and ∠BOF = 40°

Question 10. 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:

Given that OF bisects ∠BOD

∠BOF = 35°

We have to find ∠BOC and ∠AOD

∠BOD = 2 ∠BOF = 70°            [since OF bisects ∠BOD]

∠BOD = ∠AOC = 70°                 [ vertically opposite angles]

Now,

∠BOC + ∠AOC = 180°

∠BOC + 70° = 180°

∠BOC = 110°

∠AOD = ∠BOC = 110°             [Vertically opposite angles]

Hence, the value of ∠BOC = 110° and ∠AOD = 110°

Question 11. In the below figure, lines AB and CD intersect at O. If ∠AOC + ∠BOE = 70° and ∠BOD = 40°, find ∠BOE and reflex ∠COE?

Solution:

Given: AOC + BOE = 70° and BOD = 40°

We have to find ∠BOE and reflex ∠COE

BOD = AOC = 40°  [vertically opposite angles]

∠AOC + ∠BOE = 70°  [given]

⟹ 40° + ∠BOF = 70°

⟹ ∠BOF = 70° – 40°

⟹ ∠BOE = 30°

⟹ AOC + COF + BOE = 180°  [Angles in linear pair]

⟹ COE = 180° – 30° – 40°

⟹ COE = 110°

Reflex ∠COE = 360° – 110° = 250°
Hence, the value of ∠BOE = 30° and ∠COE =250°

Question 12. 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 acute angles.

(iv) If angles forming a linear pair are equal, then each of the angles have a measure of 90°

Solution:

(i) True

(ii) False

(iii) False

(iv) true

Question 13. Fill in Inc 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

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