R.d Sharma 2022 _mcqs for Class 9 Maths Chapter 10 - Lines And Angles

Answer:

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).

Answer:

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).

Answer:

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).

Answer:

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).

Answer:

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).

Answer:

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.

Answer:

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).

Answer:

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).

Answer:

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).

Answer:

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).

Answer:

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).

Answer:

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).

Answer:

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).

Answer:

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) .

Answer:

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).

Answer:

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).

Answer:

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).

Answer:

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).

Answer:

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).

Answer:

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).

Answer:

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).

Answer:

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.

Answer:

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).

Answer:

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).

Answer:

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) .

Answer:

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).

Answer:

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).

Answer:

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).

Answer:

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).

Answer:

Statement-2 (Reason): If two parallel lines are intersected by a transversal, then each pair of consecutive interior angles are supplementary.

Thus, Statement-2 is true.
          
Statement-1 (Assertion): In the given figure, if parallel lines l and m are intersected by a transversal n, then x = 25.

Given: Parallel lines l and m are intersected by a transversal n.

According to Statement-2, if two parallel lines are intersected by a transversal, then each pair of consecutive interior angles are supplementary.

∴ (3x + 5)° + (4x)° = 180°

⇒ (7x)° = 175°

⇒ x = 25
Thus, Statement-1 is true.
So, Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

Hence, the correct answer is option (a).

Answer:

Statement-2 (Reason): If two parallel lines are intersected by a transversal, then each pair of consecutive interior angles are equal.

If two parallel lines are intersected by a transversal, then each pair of consecutive interior angles are supplementary.

Thus, Statement-2 is false.
          
Statement-1 (Assertion): In the given figure, AB || CD, ∠BAO = 60º and ∠OCD = 110º, then ∠AOC = 50º.

Construction: Extended line CD till point E such that DE || AB.



Given: AB || CD

According to Statement-2, if two parallel lines are intersected by a transversal, then each pair of consecutive interior angles are supplementary.

∠BAO + ∠AEC = 180°
⇒ 60° + ∠AEC = 180°
⇒ ∠AEC = 120°              .....(1)

Now, ∠AEC and ∠OEC are linear pair of angles.

∴ ∠OEC + ∠AEC = 180°
⇒ ∠OEC + 120°= 180°           [From (1)]
⇒ ∠OEC = 60°              .....(2)

Now, ∠OCD and ∠OCE are linear pair of angles.

∴ ∠OEC + ∠OCE = 180°
⇒ 60° + ∠OCE = 180°           [From (2)]
⇒ ∠OCE = 70°             .....(3)

Now, in △OEC, using angle sum property, we have

∠OEC + ∠OCE + ∠EOC = 180°

⇒ 60° + 70° + ∠EOC = 180°          [From (2) and (3)]

⇒ ∠EOC = 50°

⇒ ∠AOC = ∠EOC = 50°

Thus, Statement-1 is true.

So, Statement-1 is true, Statement-2 is false.

Hence, the correct answer is option (c).

Answer:

Statement-2 (Reason): If two parallel lines are intersected by a transversal, then each pair of alternate angles are equal.

Thus, Statement-2 is true.

Statement-1 (Assertion): In the given figure, if PQ || RS, then ∠ACB = 90º.
Construction: Draw a line XY which is parallel to the lines PQ and RS passing through C.


Thus, XY || PQ || RS.

According to Statement-2, if two parallel lines are intersected by a transversal, then each pair of alternate angles are equal.

∴ ∠BCY = ∠CBR

⇒ ∠BCY = 50°                .....(1)

If two parallel lines are intersected by a transversal, then each pair of consecutive interior angles are supplementary.
            
∴ ∠CAQ + ∠ACY = 180°

⇒ 140° + ∠ACY = 180°

⇒ ∠ACY = 40°                .....(2)

Now, ∠ACB = ∠ACY + ∠BCY

⇒ ∠ACB = 40° + 50° = 90°                  [From (1) and (2)]

Thus, Statement-1 is true.
So, Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

Hence, the correct answer is option (a).

Answer:

Statement-2 (Reason): If a transversal intersects two parallel lines, then each pair of alternate angles are equal.

According to Alternate angles axiom, if a transversal intersects two parallel lines, then each pair of alternate angles are equal.

Thus, Statement-2 is true.
          
Statement-1 (Assertion): In the given figure, if AB || CD, ∠ABE = 130º and ∠ECD = 110º, then ∠BEC = 60º.

Construction: Drawing a line XY parallel to ST and passing through point E.

∠ABE + ∠BEX = 180º (Co-interior angles on the same side of transversal BE)

⇒ 130º + ∠BEX = 180º

⇒ ∠BEX = 50º                   .....(1)

Also,

∠ECD + ∠CEY = 180º (Co-interior angles on the same side of transversal SR)

⇒ 110º + ∠CEY = 180º

⇒ ∠CEY = 70º                   .....(2)

XY is a straight line. EB and EC stand on it.

∴ ∠BEX + ∠BEC + ∠CEY = 180º

⇒ 50º + ∠BEC + 70º = 180º             [From (1) and (2)]

⇒ ∠BEC = 180º − 120º = 60º

Thus, Statement-1 is true.
So, Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

Hence, the correct answer is option (b).

Answer:

Statement-2 (Reason): If a ray stands on a line, the sum of two adjacent angles formed is 180º.

According to the Linear pair axiom, if a ray stands on a line, then the sum of the two adjacent angles so formed is 180° and vice-versa.

Thus, Statement-2 is true.
          
Statement-1 (Assertion): In the given figure, if ACB is a straight line, then ∠ACD = 72º.

Given: ACB is a straight line, DC is a ray stand on the line.

According to Statement-2, if a ray stands on a line, the sum of two adjacent angles formed is 180º. 

∴ ∠ACD + ∠BCD = 180°

⇒ 2x + 3x = 180°

⇒ x = 36°

∴ ∠ACD = 2x = 72°

Thus, Statement-1 is true.
So, Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

Hence, the correct answer is option (a).

Answer:

Statement-2 (Reason): If two lines intersect, then vertically opposite angles are equal.

Thus, Statement-2 is true.

Statement-1 (Assertion): In the given figure, lines AB and CD intersect at O. If ∠AOC = 40º, then ∠BOC = 140º.

Given that, lines AB and CD intersect at O and ∠AOC = 40º.

∠AOC and ∠BOC are the angles on a straight line.

∴ ∠AOC + ∠BOC = 180°

⇒ 40° + ∠BOC = 180°

⇒ ∠BOC = 140°

Thus, Statement-1 is true.