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

### Question 1. In the below fig, lines l_{1}, and l_{2} intersect at O, forming angles as shown in the figure. If x = 45Â°. Find the values of x, y, z, and u.

**Solution:**

Given that X = 45Â°

Find: the value of Y, Z, and u

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

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

z = 180Â° – u

u = 180Â° – x

u = 180Â° – 45Â°

u = 135Â°

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

y = 180Â° – x

y =180Â° – 45Â°

y = 135Â°

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

**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

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

âˆ DOF = y = 50Â°

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

Now put the value of z and y

90Â° + 50Â° + x = 180Â°

x = 180Â° – 140Â°

x = 40Â°

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

### Question 3. In the given fig, find the values of x, y, and z.

**Solution:**

From the given figure

y = 25Â° [Vertically opposite angles are equal]

Now âˆ x +âˆ y = 180Â° [Linear pair of angles]

x = 180Â° – 25Â°

x = 155Â°

Also,

z = x = 155Â° [Vertically opposite angles]

y = 25Â°

z = 155Â°

So, x = 155Â°, y = 25Â°, z = 155Â°

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

**Solution:**

From the figure

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

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

3x + 5x + 2x = 180Â°

10x = 180Â°

x = 180Â°/10

x = 18Â°

So, the value of x = 18Â°

### 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

âˆ AOC = âˆ BOD [vertically opposite angles are equal] â€¦(1)

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

To Prove: POQ is a straight line.

âˆ AOP = âˆ COP [OP is the bisector of âˆ AOC]â€¦(2)

âˆ BOQ = âˆ QOD [OQ is the bisector of âˆ BOD]â€¦(3)

Now,

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

âˆ BOQ + âˆ QOD + âˆ DOA + âˆ AOP + âˆ POC + âˆ COB = 360Â°

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

âˆ QOD + âˆ DOA + âˆ AOP = 180Â°

POQ = 180Â°

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 = 360Â° [sum of all angles around a point is 360 degrees ]

2AOP + AOD + 2D0Q + BOC = 360Â°

2AOP + 2AOD + 2DOQ = 360Â°

2(AOP + AOD + DOQ) = 360Â°

AOP + AOD +DOQ = 360Â°/2

AOP + AOD + DOQ = I80Â°

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 âˆ BOC = 90Â°, âˆ AOC = 90 Â°âˆ AOD = 90Â° and âˆ BOD = 90Â°

Given:âˆ BOC = 90Â°

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

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

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

âˆ AOC = 90Â°

âˆ AOC = âˆ BOD = 90Â° [Vertically opposite angles]

Hence, âˆ AOC = âˆ BOC = âˆ BOD = âˆ AOD = 90Â°

### Question 8. In the below fig. rays AB and CD intersect at O.

**(i) Determine y when x = 60**Â°

**(ii) Determine x when y = 40**Â°

**Solution:**

(i)Given that x = 60Â°âˆ AOC + âˆ BOC = 180Â° [linear pair of angles]

âŸ¹ 2x + y = 180Â°

âŸ¹ 2(60Â°) + y = 180Â° [since x = 60Â°]

âŸ¹ y = 60Â°

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

(ii)Given y = 40Â°âˆ AOC + âˆ BOC = 180Â° [linear pair of angles]

âŸ¹ 2x + y = 180Â°

âŸ¹ 2x + 40Â° = 180Â° [since x = 40Â°]

âŸ¹ 2x =180Â° – 140Â°

âŸ¹ 2x = 140Â°

âŸ¹ x = 70Â°

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

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

**Solution:**

From the figure

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

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

âŸ¹ âˆ DOE = 180Â° – 40Â° – 35Â° = 105Â°

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

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

âˆ AOE + âˆ AOC + âˆ COF = 180Â°

âŸ¹ 40Â° + âˆ AOC +105Â° = 180Â°

âŸ¹ âˆ AOC = 180Â° – 145Â°

âŸ¹ âˆ AOC = 35Â°

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

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

### Question 10. AB, CD, and EF are three concurrent lines passing through the point O such that OF bisects BOD. If âˆ BOF = 35. Find âˆ BOC and âˆ AOD.

**Solution:**

Given that OF bisects âˆ BOD

âˆ BOF = 35Â°

We have to find âˆ BOC and âˆ AOD

âˆ BOD = 2 âˆ BOF = 70Â° [since OF bisects âˆ BOD]

âˆ BOD = âˆ AOC = 70Â° [ vertically opposite angles]

Now,

âˆ BOC + âˆ AOC = 180Â°

âˆ BOC + 70Â° = 180Â°

âˆ BOC = 110Â°

âˆ AOD = âˆ BOC = 110Â° [Vertically opposite angles]

Hence, the value of âˆ BOC = 110Â° and âˆ AOD = 110Â°

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

**Solution:**

Given: AOC + BOE = 70Â° and BOD = 40Â°

We have to find âˆ BOE and reflex âˆ COE

BOD = AOC = 40Â° [vertically opposite angles]

âˆ AOC + âˆ BOE = 70Â° [given]

âŸ¹ 40Â° + âˆ BOF = 70Â°

âŸ¹ âˆ BOF = 70Â° – 40Â°

âŸ¹ âˆ BOE = 30Â°

âŸ¹ AOC + COF + BOE = 180Â° [Angles in linear pair]

âŸ¹ COE = 180Â° – 30Â° – 40Â°

âŸ¹ COE = 110Â°

Reflex âˆ COE = 360Â° – 110Â° = 250Â°

Hence, the value of âˆ BOE = 30Â° and âˆ COE =250Â°

### 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 90Â°**

**(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 90Â°**

**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 180Â°, then the ______ arms of the two angles are opposite rays.**

**Solution:**

(i)Obtuse angle

(ii)180Â°

(iii)Uncommon

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