RD Sharma 2020 solution class 9 chapter 10 Lines and Angles MCQS

MCQS

Page-10.50

Question 1:

One angle is equal to three times its supplement. The measure of the angle is

(a) 130°

(b) 135°

(c) 90°

(d) 120°

Answer 1:

Let the supplement of the angle be

Therefore, according to the given statement, the required angle measures

Since the angles are supplementary, therefore their sum must be equal to

Or we can say that

Thus, the supplement of angle measures

Hence, the correct choice is (b).

Question 2:

Two straight line AB and CD intersect one another at the point O. If ∠AOC + ∠COB + ∠BOD = 274°, then ∠AOD =

(i) 86°

(ii) 90°

(iii) 94°

(iv) 137°

Answer 2:

Let us draw the following diagram showing two linesand intersecting at a point.

Thus, $\angle AOD,\angle AOC,\angle COB \mathrm{and} \angle BOD$ form a complete angle, that is the sum of these four angle is.

That is,

$\angle AOD+\angle AOC+\angle COB+\angle BOD=360°$                    ... (i)

It is given that

                               ...(ii)

Subtracting (ii) from (i), we get:

Hence, the correct choice is (a).

Question 3:

Two straight lines AB and CD cut each other at O. If ∠BOD = 63°, then ∠BOC =

(a) 63°

(b) 117°

(c) 17°

(d) 153°

Answer 3:

Let us draw the following diagram showing two linesand intersecting each other at a point.

Let the required angle measures.

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

That is,

It is given that. Substituting, this value above, we get:

Hence, the correct choice is (b).

Question 4:

Consider the following statements:
When two straight lines intersect:

(i) adjacent angles are complementary

(ii) adjacent angles are supplementary

(iii) opposite angles are equal

(iv) opposite angles are supplementary

Of these statements
 
(a) (i) and (ii) are correct

(b) (ii) and (iii) are correct

(c) (i) and (iv) are correct

(d) (ii) and (iv)  are correct

Answer 4:

Let us draw the following diagram showing two straight lines AD and BC intersecting each other at a point.

Now, let us consider each statement one by one:

(i)

When two lines intersect adjacent angles are complementary.

This statement is incorrect

Explanation:

As the adjacent angles form a linear pair and they are supplementary.

(ii)

When two lines intersect adjacent angles are supplementary.

This statement is correct.

Explanation:

As the adjacent angles form a linear pair and they are supplementary.

(iii)

When two lines intersect opposite angles are equal.

This statement is correct.

Explanation:

As the vertically opposite angles are equal.

(iv) When two lines intersect opposite angles are supplementary.

This statement is incorrect.

Explanation:

As the vertically opposite angles are equal

Thus, out of all, (ii) and (iii) are correct.

Hence, the correct choice is (b).

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Question 5:

Given ∠POR = 3x and ∠QOR = 2x + 10°. If POQ is a straight line, then the value of x is

(a) 30°

(b) 34°

(c) 36°

(d) none of these

Answer 5:

Let us draw the following figure, showingas a straight line.

Thus, and form a linear pair, therefore their sum must be supplementary. That is;

It is given that

and

On substituting these two values above, we get:

Hence, the correct choice is (b).

Question 6:

In the given figure, AOB is a straight line. If ∠AOC + ∠BOD = 85°, then ∠COD =

(a) 85°

(b) 90°

(c) 95°

(d) 100°

Answer 6:

It is given that is a straight line.

Also,,and form a linear pair.

Therefore, their sum must be supplementary.

That is

   ...(i)

It is given that

$\angle AOC+\angle BOD=85°$                      ...(ii)

On substituting the value of (ii) in (i) we get,

$$\begin{gathered} \angle COD+85°=180° \\ \Rightarrow \angle COD=180°-85° \\ \Rightarrow \angle COD=95° \end{gathered}$$
Hence, (c) is the correct option.

 

Question 7:

In the given figure, the value of y is

(a) 20°

(b) 30°

(c) 45°

(d) 60°

Answer 7:

In the given figure,and are vertically opposite angles, therefore, these must be equal.

That is,

          ...(i)

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

That is,

From (i) equation, we get:

From (i) equation again,

Hence, the correct choice is (b).

Question 8:

In the given figure, the value of x is

(a) 12

(b) 15

(c) 20

(d) 30

Answer 8:

The figure is as follows:

It is given that

Also,

         (vertically opposite angles)

Since, x°, and form a linear pair.

Therefore,

Hence, the correct choice is (c).

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Question 9:

In the given figure, which of the following statements must be true?
(i) a + b = d + c

(ii) a + c + e = 180°

(iii) b + f = c + e

(a) (i) only

(b) (ii) only

(c) (iii) only

(d) (ii) and (iii) only

Answer 9:

Now, let us consider each statement one by one:

(i)

Statement:

This statement is incorrect

Explanation:

We have, a and d are vertically opposite angles.

Therefore,

(I)

Similarly, b and e are vertically opposite angles.

Therefore,

(II)

On adding (I) and (II), we get:

Thus, this statement is incorrect.

(ii)

Statement:

This statement is correct.

Explanation:

As , and form a linear pair, therefore their sum must be supplementary.

(III)

Also andare vertically opposite angles, therefore, these must be equal.

Putting in (III), we get:

(iii)

Statement:

This statement is correct.’

Explanation:

As, and form a linear pair, therefore their sum must be supplementary.

(IV)

Also , and form a linear pair, therefore their sum must be supplementary.

(V)

On comparing (IV) and (V), we get:

Also andare vertically opposite angles, therefore, these must be equal.

Therefore,

Substituting the above equation in (VI), we get:

Thus, out of all, (ii) and (iii) are correct.

Hence, the correct choice is (d).

Question 10:

If two interior angles on the same side of a transversal intersecting two parallel lines are in the ratio 2:3, then the measure of the larger angle is

(a) 54°

(b) 120°

(c) 108°

(d) 136°

Answer 10:

Let us draw the following figure:

Here with t as a transversal.

Also, and are the two angles on the same side of the transversal.

It is given that

Therefore, let

and

We also, know that, if a transversal intersects two parallel lines, then each pair of consecutive interior angles are supplementary.

Therefore,

On substituting andin equation above, we get:

 

Clearly,

Therefore,

Also,

Hence, the correct choice is (c).

Question 11:

In the given figure, if AB || CD, then the value of x is

(a) 20°

(b) 30°

(c) 45°

(d) 60°

Answer 11:

Here

Also, ∠1 andare the two corresponding angles.

Then, according to the Corresponding Angles Axiom, which states:

If a transversal intersects two parallel lines, then each pair of corresponding angles are equal.

Therefore,

Also,and form a linear pair, therefore, their sum must be supplementary.

Therefore,

On substituting in equation above, we get:

Hence, the correct choice is (b).

Question 12:

Two lines AB and CD intersect at O. If ∠AOC + ∠COB + ∠BOD = 270°, then ∠AOC =

(a) 70°

(b) 80°

(c) 90°

(d) 180°

Answer 12:

Let us draw the following diagram showing two linesand intersecting at a point.

 

Thus,,, and form a complete angle, that is the sum of these four angle is .

That is,

(I)

It is given that

(II)

Subtracting (II) from (I), we get:

If one of the four angles formed by two intersecting lines is a right angle, then each of the four angles will be a right angle.
So, ∠AOC = $90°$

Hence, the correct choice is (c).

Question 13:

In the given figure, PQ || RS, ∠AEF = 95°, ∠BHS = 110° and ∠ABC = x°. Then the value of x is

(a) 15°

(b) 25°

(c) 70°

(d) 35°

Answer 13:

In the given figure,

.

 

Also,and are the corresponding angles.

Then, according to the Corresponding Angles Axiom, which states:

If a transversal intersects two parallel lines, then each pair of corresponding angles are equal.

Therefore,

It is given that

Therefore,

Clearly, and form a linear pair, therefore, their sum must be supplementary.

Therefore,

On substituting in equation above, we get:

In ΔBHG:

We know that, in a triangle exterior angle is equal to the sum of the interior opposite angles. Therefore,

Substituting

and , we get :

Hence the correct choice is (b).

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Question 14:

In the given figure, if l₁ || l₂, what is the value of x?

Answer 14:

In the given figure:

Since, therefore, the pair of corresponding angles should be equal.

That is;

Also, $\angle 1 \mathrm{and} \angle 2$ are vertically opposite angles, therefore,

Since 58°, ∠2 and x form a linear pair. Therefore,

Hence, the correct choice is (b) .

Question 15:

In the given figure, if l₁ || l₂, what is x + y in terms of w and z?

(a) 180 − w + z

(b) 180 + w − z

(c) 180 − w − z

(d) 180 + w + z

Answer 15:

The figure is given below:

 

Since, y and z are alternate interior opposite angles. Therefore, these must be equal.

(i)

Also x and w are consecutive interior angles.

Theorem states: If a transversal intersects two parallel lines, then each pair of consecutive interior angles are supplementary.

Therefore,

(ii)

On adding equation (i) and (iii) , we get :

Hence, the correct choice is (a).

Question 16:

In the given figure, if l₁ || l₂, what is the value of y?



(a) 100

(b) 120

(c) 135

(d) 150

Answer 16:

Given figure is as follows:

It is given that .

and 3x are vertically opposite angles, which must be equal, that is,

(i)

Also, and x are consecutive interior angles.

Theorem states: If a transversal intersects two parallel lines, then each pair of consecutive interior angles are supplementary.

Thus,

From equation (i), we get:

x and y form a linear pair. Therefore, their sum must be supplementary.

Thus,

Substituting, in equation above, we get:

Hence, the correct choice is (c).

Page-10.54

Question 17:

In the given figure, if l₁ || l₂ and l₃ || l₄, what is y in terms of x?

(a) 90 + x

(b) 90 + 2x

(c) $90-\frac{x}{2}$

(d) 90 − 2x

Answer 17:

The given figure is:

Here, we have ∠2 and 2y are vertically opposite angles. Therefore,

       ...(i)

and x are alternate interior opposite angles.

Thus,

         ...(ii)

and are consecutive interior angles.

Theorem states: If a transversal intersects two parallel lines, then each pair of consecutive interior angles are supplementary.

Thus,

From (i) and (ii), we get:

Hence, the correct choice is (c).

Question 18:

In the given figure, if l || m, what is the value of x?



(a) 60

(b) 50

(c) 45

(d) 30

Answer 18:

Given figure is as follows:

Sinceand are vertically opposite angles, therefore,

Also, 3y and are alternate interior opposite angles, therefore,

Substituting in equation (i), we get:

Hence the correct choice is (a).

Question 19:

In the given figure, if AB || HF and DE || FG, then the measure of ∠FDE is

(a) 108°

(b) 80°

(c) 100°

(d) 90°

Answer 19:

The given figure is as follows:

It is given that .

Thus, x and ∠HFC form a linear pair, therefore,

Also

Thus, x and ∠FDB are corresponding interior opposite angles, therefore,

From (i):

Thus,

Hence, the correct choice is (b).

Question 20:

In the given figure, if lines l and m are parallel, then x =

(a) 20°

(b) 45°

(c) 65°

(d) 85°

Answer 20:

The given figure is as follows:

Since, . Thus, angle and ∠1 are corresponding angles.

Therefore,

(i)

In a triangle, we know that, the exterior angle is equal to the sum of the interior opposite angle.

In ΔAOB:

From equation (i):

Hence, the correct choice is (b).

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Question 21:

In the given figure, if AB || CD, then x =

(a) 100°

(b) 105°

(c) 110°

(d) 115°

Answer 21:

The given figure is as follows:

It is given that .

Let us draw a line PQ parallel to AB and CD.

It is given that,

(i)

Since, . Thus, angle and ∠1 are consecutive interior angles.

Therefore,

Similarly, . Thus, x angle and ∠2 are corresponding angles.

Therefore,

(iii)

On substituting (ii) and (iii) in (i):

Hence, the correct choice is (a).

Question 22:

In the given figure, if lines l and m are parallel lines, then x =

(a) 70°

(b) 100°

(c) 40°

(d) 30°

Answer 22:

We have the following figure:

It is given that

We know that consecutive interior angles are supplementary.

Therefore,

$\angle 1=\angle AOB=110 (\mathrm{vertically} \mathrm{opposite} \mathrm{angles})$

In a triangle, we know that, the sum of the angles is supplementary.

In ΔAOB:

$$\begin{gathered} 30°+x+110°=180° \\ \Rightarrow x=180-110-30=40 \end{gathered}$$

Hence, the value of x will be $40°$.
Thus, (c) is the correct answer.

Question 23:

In the given figure, if l || m, then x =

(a) 105°

(b) 65°

(c) 40°

(d) 25°

Answer 23:

The given figure:

Let us draw a line n parallel to l and m.

Thus, we can say that .

Also, from the figure we get :

                ...(i)

Since .

Thus, alternate interior opposite angles are equal. That is,

                     ...(ii)

Since .

Thus, alternate interior opposite angles are equal. That is,

                    ...(iii)

On substituting, equation (ii) and (iii) in (i):

Hence, the correct choice is (a).

Question 24:

In the given figure, if lines l and m are parallel, then the value of x is

(a) 35°

(b) 55°

(c) 65°

(d) 75°

Answer 24:

The given figure is as follows with :

Also, ∠1 and ∠2 form a linear pair. Thus,

It is given that ∠2 = 90°, substituting this value , we get :

In a triangle, we know that, the exterior angle is equal to the sum of the interior opposite angle.

In ΔAOB:

From equation (i):

Hence, the correct choice is (a).

Page-10.56

Question 25:

Two complementary angles are such that two times the measure of one is equal to three times the measure of the other. The measure of the smaller angle is

(a) 45°

(b) 30°

(c) 36°

(d) none of these

Answer 25:

Let one angle be.

Then, the other complementary angle becomes

It is given that two times the angle measuringis equal to three times the angle measuring

Or, we can say that:

On dividing both sides of the equation by 5,we get

Also, the other complementary angle becomes

Thus, the measure of the required smaller angle is.

Hence, the correct choice is (c) .

Question 26:

In the given figure, if $\frac{y}{x} = 5$and $\frac{z}{x} = 4$, then the value of x is

(a) 8°

(b) 18°

(c) 12°

(d) 15°

Answer 26:

In the given figure, we have,and forming a linear pair, therefore these must be supplementary.

That is,

                        (i)

Also,

And

Substituting (ii) and (iii) in (i), we get:

Hence, the correct choice is (b).

Question 27:

In the given figure, AB || CD || EF and GH || KL. The measure of ∠HKL is

(a) 85°

(b) 135°

(c) 145°

(d) 215°

Answer 27:

The given figure is as follows:

Let us extend GH to meet AB at Y.

We have the following:

and are the supplementary angles. Therefore,

Since,. Thus, and are the interior alternate angles.

Therefore,

Since, . Thus, and are the corresponding angles.

Therefore,

From (1), we get:

……(3)

Since . Thus are corresponding angles,

Therefore,

Thus, the required angle x can be calculated as:

From (3) and (4) we get:

Hence, the correct choice is (c).

Question 28:

AB and CD are two parallel lines. PQ cuts AB and CD at E and F respectively. EL is the bisector of ∠FEB. If ∠LEB = 35°, then ∠CFQ will be

(a) 55°

(b) 70°

(c) 110°

(d) 130°

Answer 28:

The figure is given as follows:

It is given that,with PQ as transversal.

Also, EL is the bisector and.

We need to find.

Since, EL is the bisector and.

Therefore,

We have , the and are consecutive interior angles, which must be supplementary.

From equation (i), we get:

We have and as vertically opposite angles.

Therefore,

Hence, the correct choice is (c).

Question 29:

In the given figure, If line segment AB is parallel to the line segment CD, what is the value of y?

(a) 12

(b) 15

(c) 18

(d) 20

Answer 29:

The figure is given as follows:

It is given that AB is parallel to CD.

Thus,and $\angle BDC$ are consecutive interior angles.

Therefore, their sum must be supplementary.

That is,

$\angle ABD+\angle BDC=180°$

From the figure, we get:

Hence, the correct choice is (d).

Page-10.57

Question 30:

In the given figure, if CP || DQ, then the measure of x is

(a) 130°

(b) 105°

(c) 175°

(d) 125°

Answer 30:

Let us extend PC to meet AB at point O.

It is given that .

Thus,and are corresponding angles. Therefore,

Given that, then we have:

Or,

Also, in ΔAOC, exterior angle is equal to the sum of the interior opposite angles, therefore,

Hence, the correct choice is (a).