RD Sharma 2020 solution class 9 chapter 10 Lines and Angles Exercise 10.3

Exercise 10.3

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Question 1:

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

Answer 1:

It is given that lines and intersect at a point.

Therefore,and are the two linear pairs are formed.

Thus,

Also,

It is given that, putting this value above, we get:

Also we have a two pairs of vertically opposite angles in the figure, that is,and .

We know that, if two lines intersect, then the vertically opposite angles are equal.

Thus,

And

Question 2:

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

Answer 2:

It is given that the lines, and intersect at a point.

Therefore, vertically opposite angles should be equal

Also, ,and form a linear pair.

Therefore,

Substituting ,and in equation above:

andare vertically opposite angles.

Therefore,

Substituting , in equation above:

Similarly, ,are vertically opposite angles.

Therefore,

Substituting , in equation above:

Question 3:

In the given figure, find the values of x, y and z.

Answer 3:

In the given question, the values of x, y, and z will be determined as follows:
z and $25°$ form a linear pair.
$$\begin{gathered} \mathrm{So}, z + 25°=180° \\ \Rightarrow z=180-25 \\ \Rightarrow z=155° \end{gathered}$$

Now, z and x are vertically opposite to each other. So, x = $155°$.

Also, y and x form a linear pair. 
$$\begin{gathered} \mathrm{So}, y+ 155°=180° \\ \Rightarrow y=180-155 \\ \Rightarrow y=25° \end{gathered}$$

Hence, the values are $x=155°, y=25° \mathrm{and} z=155°$.

Question 4:

In the given figure, find the value of x.

Answer 4:

In the following figure we have to find the value of x

In the figure AB, CD and EF are lines; therefore, angles COF and EOD are vertically opposite angles.

Therefore,

Since, AB is a straight line, so

Hence, .

Question 5:

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.

Answer 5:

The given problem can be drawn as :

It is given that

Also,and form a linear pair.

Therefore, their sum must be equal to.

Substituting, above, we get:

Similarly, we can prove that

and

Hence, we have proved that ,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.

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Question 6:

In the given figure, rays AB and CD intersect at O.


(i) Determine y when x = 60°

(ii) Determine x when y = 40

Answer 6:

Raysand intersect at point.

Therefore, and form a linear pair.

Thus,

(i)

On substituting:

(ii)

On substituting:

Question 7:

In the given figure, lines AB, CD and EF intersect at O, Find the measure of ∠AOC, ∠COF, ∠DOE and ∠BOF.

Answer 7:

It is given thatand intersect at a point

Thus and are vertically opposite angles, therefore, these must be equal.

That is,

Similarly,and intersect at a point.

Thusand are vertically opposite angles, therefore, these must be equal.

That is,

Similarly,and intersect at a point.

Thusand are vertically opposite angles, therefore, these must be equal.

That is,

Also,,and form a linear pair. Therefore, their sum must be equal to .

Putting in (I):

Question 8:

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.

Answer 8:

The corresponding figure is as follows:

Three concurrent lines are given as follows:

AB,CD and EF

Also, OF is the bisector of and it is given that.Therefore,

Also,

Since, andare vertically opposite angles. Therefore,

From (i) equation:

We know that and form a linear pair.

Thus,

Similarly, and form a linear pair.

Thus,

Question 9:

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

Answer 9:

In the figure, ,and form a linear pair.

Thus,

It is given that, on substituting this value, we get:

Thus, reflex

Therefore, reflex

Sinceand are vertically opposite angles, thus, these two must be equal.

Therefore,

But, it is given that :

Substituting in above equation:

Question 10:

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, then each angle measures 90°.
(iii) Angles forming a linear pair can both be acute angles.
(iv) If angles forming a linear pair are equal, then each of these angles is of measure 90°.

Answer 10:

(i) True

As the sum of the angles forming a linear pair is.

(ii) False

As the statement is incomplete in itself.

(iii) False

Let us assume one of the angle in a linear pair be; such that ,that is, an acute angle.

Therefore, the other angle in the linear pair becomes, which clearly cannot be acute.

(iv) True

Let one of the angle in the linear pair be. Then, other angle also becomes equal to.

Therefore, by the definition of linear pair, we get:

.

Hence, if angles forming a linear pair are equal, then each of these angles is of measure.

Question 11:

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.

Answer 11:

(i)

If one angle of a linear pair be acute, then its other angle will be obtuse.

Explanation:

Let us assume one of the angle in a linear pair be; such that,that is, an acute angle.

Therefore, the other angle in the linear pair becomes, which clearly cannot be acute.

(ii)

A ray stands on a line, and then the sum of the two adjacent angles so formed is.

Explanation:

The statement talks about two adjacent angles forming a linear pair.

(iii) If the sum of the two adjacent angles is, then the uncommon arms of the two angles are opposite rays.

Explanation:

The statement talks about two adjacent angles forming a linear pair.

Therefore, this can be drawn diagrammatically as:

Question 12:

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

Answer 12:

Let AB and CD intersect at a point O

Also, let us draw the bisectors OP and OQ of and.

Therefore,

And

We know that,and are vertically opposite angles. Therefore, these must be equal, that is:

We know that:

From (i)

From (ii)

This means, , and form a linear pair.

Hence, POQ forms a straight line.

Thus, we can say that the bisectors of a pair of vertically opposite angles are in the same straight line.

Question 13:

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.

Answer 13:

Let AB and CD intersect at a point O

Also, let us draw the bisector OP of .

Therefore,

Also, let’s extend OP to Q.

We need to show that, OQ bisects.

Let us assume that OQ bisects, now we shall prove that POQ is a line.

We know that,

and are vertically opposite angles. Therefore, these must be equal, that is:

and are vertically opposite angles. Therefore,

Similarly,

We know that:

Thus, POQ is a straight line.

Hence our assumption is correct. That is,

We can say that if the two straight lines intersect each other, then the ray opposite to the bisector of one of the angles thus formed bisects the vertically opposite angles.