RD Sharma 2020 solution class 9 chapter 10 Lines and Angles Exercise 10.4

Exercise 10.4

Page-10.46

Question 1:

In the given figure, AB CD and ∠1 and ∠2 are in the ratio 3:2 Determine all angles form 1 to 8.

Answer 1:

The given figure is as follows:

It is give that the lines AB and CD are parallel and angles 1 and 2 are in the ratio 3: 2.

Let

In the figure angle 1 and 2 are supplementary. So,

3x + 2x = 180

⇒ 5x = 180

⇒ x = 36
$\angle 1=36\times 3=108° \mathrm{and} \angle 2=36\times 2=72°$

Since, angle 1 and 5 and angle 2 and 6 are corresponding angles, so

Since, angles 1 and 3 and 2 and 4 are vertically opposite angles, so

Now,

Angle 5 and 6 and angle 6 and 8 are vertically opposite angles, so

Hence,and.

Question 2:

In the given figure, l, m and n are parallel lines intersected by transversal p at X, Y and Z respectively. Find ∠1, ∠2 and ∠3.

Answer 2:

According to the given figure,
m || n and are cut by transversal p.  
$\angle 2=120° (\mathrm{alternate} \mathrm{interior} \mathrm{angles} \mathrm{are} \mathrm{equal})$
Also, l || m. So, $\angle 1=\angle 3 (\mathrm{corresponding} \mathrm{angles})$
Also, $\angle 3 \mathrm{and} 120° \mathrm{form} \mathrm{a} \mathrm{linear} \mathrm{pair}.$
$$\begin{gathered} \angle 3+120°=180° \\ \Rightarrow \angle 3=180-120 \\ \Rightarrow \angle 3=60° \end{gathered}$$

And $\angle 1=\angle 3=60°,\angle 2=120°$
 

Question 3:

In the given figure, if AB || CD and CD || EF, find ∠ACE.

Answer 3:

The figure is given as follows:

It is given that AB || CD and CD || EF

Thus,and are alternate interior opposite angles.

Therefore,

Also, we have

From the figure:

From equations (i) and (ii):

Hence, the required value for is.

Question 4:

In the given figure, state which lines are parallel and why.

Answer 4:

The given figure is as follows:

Since

These are the pair of alternate interior opposite angles.

Theorem states: If a transversal intersects two lines in such a way that a pair of alternate interior angles is equal, then the two lines are parallel.

Therefore,

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Question 5:

In the given figure, if l || m, n || p and ∠1 = 85°, find ∠2.

Answer 5:

The figure is given as follows:

It is given that .

Thus,and are corresponding angles.

Therefore,

It is given that . Therefore,

                ...(i)

Also, we have .

Thus,and are consecutive interior angles.

Therefore,

From equation (i), we get:

Hence, the required value for is .

Question 6:

If two straight lines are perpendicular to the same line, prove that they are parallel to each other.

Answer 6:

The figure can be drawn as follows:

 

Here, and.

We need to prove that

It is given that , therefore,

(i)

Similarly, we have , therefore,

(ii)

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

But these are the pair of corresponding angles.

Theorem states: If a transversal intersects two lines in such a way that a pair of corresponding angles is equal, then the two lines are parallel.

Thus, .

Question 7:

Two unequal angles of a parallelogram are in the ratio 2 : 3. Find all its angles in degrees.

Answer 7:

The parallelogram can be drawn as follows:

It is given that

Therefore, let:

and

We know that opposite angles of a parallelogram are equal.

Therefore,

Similarly

Also, if , then sum of consecutive interior angles is equal to .

Therefore,

We have

Also,

Similarly,

And

Hence, the four angles of the parallelogram are as follows:

, , and .

Question 8:

In each of the two lines is perpendicular to the same line, what kind of lines are they to each other?

Answer 8:

The figure can be drawn as follows:

Here,and.

We need to find the relation between lines l and m

It is given that , therefore,

(i)

Similarly, we have, therefore,

(ii)

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

But these are the pair of corresponding angles.

Theorem states: If a transversal intersects two lines in such a way that a pair of corresponding angles is equal, then the two lines are parallel.

Thus, we can say that .

Hence, the lines are parallel to each other.

Question 9:

In the given figure, ∠1 = 60° and ∠2 = ${\left(\frac{2}{3}\right)}^{rd}$ of a right angle. Prove that l || m.

Answer 9:

The figure is given as follow:

 

It is given that

Also,

Thus we have

But these are the pair of corresponding angles.

Thus

Hence proved.

Question 10:

In the given figure, if l || m || n and  ∠1 = 60°, find ∠2.

Answer 10:

The given figure is as follows:

 

We have and

Thus, we get and as corresponding angles.

Therefore,

(i)

We haveand forming a linear pair.

Therefore, they must be supplementary. That is;

From equation (i):

(ii)

We have

Thus, we get and as alternate interior opposite angles.

Therefore, these must be equal. That is,

From equation (ii), we get :

Hence the required value for is .

Question 11:

Prove that the straight lines perpendicular to the same straight line are parallel to one another.
 

Answer 11:

The figure can be drawn as follows:

Here, and.

We need to prove that

It is given that , therefore,

(i)

Similarly, we have, therefore,

(ii)

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

But these are the pair of corresponding angles.

Theorem states: If a transversal intersects two lines in such a way that a pair of corresponding angles is equal, then the two lines are parallel.

Thus, we can say that .

Question 12:

The opposite sides of a quadrilateral are parallel. If one angle of the quadrilateral is 60°, find the other angles.

Answer 12:

The quadrilateral can be drawn as follows:

Here, we have and.

Also,.

Since,.Thus, and are consecutive interior angles.

Thus these two must be supplementary. That is,

Similarly, .Thus,and are consecutive interior angles.

Thus these two must be supplementary. That is,

Similarly,.Thus,and are consecutive interior angles.

Thus these two must be supplementary. That is,

Hence the other angles are as follows:

Question 13:

Two lines AB and CD intersect at O. If ∠AOC + ∠COB + ∠BOD = 270°, find the measures of ∠AOC, ∠COB, ∠BOD and ∠DOA.

Answer 13:

Since, lines AB and CD intersect each other at point O.

Thus,and are vertically opposite angles.

Therefore,

…… (I)

Similarly,

…... (II)

 

Also, we have ,,and forming a complete angle. Thus,

It is given that

Thus, we get

From (II), we get:

We know thatand form a linear pair. Therefore, these must be supplementary.

From (I), we get:

Question 14:

In the given figure, p is a transversal to lines m and n, ∠2 = 120° and ∠5 = 60°. Prove that m || n.

Answer 14:

The figure is given as follows:

It is given that p is a transversal to lines m and n .Also,

and .

We need to prove that

We have .

Also,and are vertically opposite angles, thus, these two must be equal. That is,

(i)

Also,.

Adding this equation to (i), we get :

But these are the consecutive interior angles.

Theorem states: If a transversal intersects two lines in such a way that a pair of consecutive interior angles is supplementary, then the two lines are parallel.

Thus, .

Hence, the lines are parallel to each other.

Question 15:

In the given figure, transversal l intersects two lines m and n, ∠4 = 110° and ∠7 = 65°. Is m || n ?

Answer 15:

The figure is given as follows:

It is given that l is a transversal to lines m and n. Also,

and .

We need check whether or not.

We have.

Also,and are vertically opposite angles, thus, these two must be equal. That is,

(i)

Also,.

Adding this equation to (i), we get:

But these are the consecutive interior angles which are not supplementary.

Theorem states: If a transversal intersects two lines in such a way that a pair of consecutive interior angles is supplementary, then the two lines are parallel.

Thus, m is not parallel to n.

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Question 16:

Which pair of lines in the given figure are parallel? Given reasons.

Answer 16:

The figure is given as follows:

We haveand.

Clearly,

.

These are the pair of consecutive interior angles.

Theorem states: If a transversal intersects two lines in such a way that a pair of consecutive interior angles is supplementary, then the two lines are parallel.

Thus, .

Similarly, we have and.

Clearly,

.

These are the pair of consecutive interior angles.

Theorem states: If a transversal intersects two lines in such a way that a pair of consecutive interior angles is supplementary, then the two lines are parallel.

Thus,.

Hence the lines which are parallel are as follows:

and .

Question 17:

If l, m, n are three lines such that l || m and n $\perp$ l. prove that n $\perp$ m.

Answer 17:

The figure can be drawn as follows:

 

 

Here, and

We need to prove that .

It is given that, therefore,

(i)

We have, thus,and are the corresponding angles. Therefore,these must be equal. That is,

From equation (i), we get:

Therefore,.

Hence proved.

Question 18:

Which of the following statements are true (T) and which are false (F)? Give reasons.

(i) If two lines are intersected by a transversal, then corresponding angles are equal.
(ii) If two parallel lines are intersected by a transversal, then alternate interior angles are equal.
(iii) Two lines perpendicular to the same line are perpendicular to each other.
(iv) Two line parallel to the same line are parallel to each other.
(v) If two parallel lines are intersected by a transversal, then the interior angles on the same side of the transversal are equal.

Answer 18:

(i)

Statement: If two lines are intersected by a transversal, then corresponding angles are equal.

False

Reason:

The above statement holds good if the lines are parallel only.

(ii)

Statement: If two parallel lines are intersected by a transversal, then alternate interior angles are equal.

True

Reason:

Let l and m are two parallel lines.

And transversal t intersects l and m making two pair of alternate interior angles, ,and,.

 

We need to prove that and .

We have,

(Vertically opposite angles)

And, (corresponding angles)

Therefore,

(Vertically opposite angles)

Again, (corresponding angles)

Hence, and .

(iii)

Statement: Two lines perpendicular to the same line are perpendicular to each other.

False

Reason:

The figure can be drawn as follows:

Here, and

It is given that , therefore,

(i)

Similarly, we have , therefore,

(ii)

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

But these are the pair of corresponding angles.

Theorem states: If a transversal intersects two lines in such a way that a pair of corresponding angles is equal, then the two lines are parallel.

Thus, we can say that .

(iv)

Statement: Two lines parallel to the same line are parallel to each other.

True

Reason:

The figure is given as follows:

 

It is given that and

We need to show that

We have , thus, corresponding angles should be equal.

That is,

Similarly,

Therefore,

But these are the pair of corresponding angles.

Therefore, .

(v)

Statement: If two parallel lines are intersected by a transversal, then interior angles on the same side of the transversal are equal.

False

Reason:

Theorem states: If a transversal intersects two parallel lines then the pair of alternate interior angles is equal.

Question 19:

Fill in the blanks in each of the following to make the statement true:

(i) If two parallel lines are intersected by a transversal, then each pair of corresponding angles are ...
(ii) If two parallel lines are intersected by a transversal, then interior angles on the same side of the transversal are ....
(iii) Two lines perpendicular to the same line are ... to each other.
(iv) Two lines parallel to the same line are ... to each other.
(v) If a transversal intersects a pair of lines in such away that a pair of alternate angles are equal, then the lines are ...
(vi) If a transversal intersects a pair of lines in such away that the sum of interior angles on the same side of transversal is 180°, then the lines are ...

Answer 19:

(i) If two parallel lines are intersected by a transversal, then corresponding angles are equal.

(ii) If two parallel lines are intersected by a transversal, then interior angles on the same side of the transversal are supplementary.

(iii) Two lines perpendicular to the same line are parallel to each other.

(iv) Two lines parallel to the same line are parallel to each other.

(v) If a transversal intersects a pair of lines in such a way that a pair of interior angles is equal, then the lines are parallel.

(vi) If a transversal intersects a pair of lines in such a way that a pair of interior angles on the same side of transversal is, then the lines are parallel.

Question 20:

In the given figure, AB || CD || EF and GH || KL. Find the ∠HKL.

Answer 20:

The given figure is as follows:

 

Let us extend GH to meet AB at Y.

Similarly, extend LK to meet CD at Z.

We have the following:

and are the vertically opposite angles. Therefore,

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

Therefore,

From (i), we get:

Since,. Thus,and are the corresponding angles.

Therefore,

From (ii), we get:

(iii)

Also,and are the alternate interior opposite angles.

Therefore,

(iv)

Thus, the required angle can be calculated as:

From (iii) and (iv) we get:

Hence, the required value for is.

Question 21:

In the given figure, show that AB || EF.

Answer 21:

The figure is given as follows:

 

We need to prove that.

It is given that and

$$\begin{gathered} \angle ACD=\angle ACE+\angle ECD \\ \angle ACD=22°+35° \\ \angle ACD=57° \end{gathered}$$

Thus,

But these are the pair of alternate interior opposite angles.

Theorem states: If a transversal intersects two lines in such a way that a pair of alternate interior angles is equal, then the two lines are parallel.

Therefore,

(i)

It is given that and

Thus,

But these are the pair of consecutive interior opposite angles.

Theorem states: If a transversal intersects two lines in such a way that a pair of consecutive interior angles is supplementary, then the two lines are parallel.

Therefore,

(ii)

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

Hence proved.

Page-10.49

Question 22:

In the given figure, PQ || AB and PR || BC. If ∠QPR = 102°, determine ∠ABC. Give reasons.

Answer 22:

The figure is given as follows:

We need to find

Let us produce BA to meet PR at point G.

It is given that.

Thus, and are corresponding angles.

Therefore,

Also it is given that

(i)

Similarly, it is given that.

Thus,and are consecutive interior angles.

Therefore,

From equation (i) :

Hence, the required value for is.

Question 23:

Prove that if the two arms of an angle are perpendicular to the two arms of another angle, then the angles are either equal or supplementary.

Answer 23:

The figure is given as follows:

 

It is given that two sides AB and AC of are perpendicular to sides EF and DE of respectively.

We need to prove that either or .

It's given that , thus,

Similarly,

We know that, if opposite angles of a quadrilateral are equal, then it’s a parallelogram.

Therefore,

AMEN is a parallelogram.

Also, we know that opposite angles of a parallelogram are equal.

Therefore,

By angle sum property of a quadrilateral, we have:

Hence proved.

Question 24:

In the given figure, lines AB and CD are parallel and P is any point as shown in the figure. Show that ∠ABP + ∠ CDP = ∠DPB.

Answer 24:

The given figure is:

It is give that

Let us draw a line passing through point P and parallel to AB and CD.

We have , thus, and are alternate interior opposite angles. Therefore,

(i)

Similarly, we have, thus, and are alternate interior opposite angles. Therefore,

(ii)

On adding (i) and (ii):

Hence proved.

Question 25:

In the given figure, AB || CD and P is any point shown in the figure. Prove that:
∠ABP + ∠BPD + ∠CDP = 360°

Answer 25:

The given figure is as follows:

It is give that

Let us draw a line passing through point P and parallel to AB and CD.

We have, thus, and are consecutive interior angles. Therefore,

(i)

Similarly, we have , thus, and are consecutive interior angles. Therefore,

(ii)

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

Hence proved .

Question 26:

In the given figure, arms BA and BC of ∠ABC are respectively parallel to arms ED and EF of ∠DEF. Prove that ∠ABC = ∠DEF.

Answer 26:

The figure is given as follows:

It is given that, arms BA and BC of are respectively parallel to arms ED and EF of .

We need to show that

Let us extend BC to meet EF.

We have. and are corresponding angles, these two should be equal.

Therefore,

Hence proved.

Question 27:

In the given figure, arms BA and BC of ∠ABC are respectively parallel to arms ED and EF of ∠DEF. Prove that ∠ABC + ∠DEF = 180°

Answer 27:

The figure is given as follows:

It is given that, arms BA and BC of are respectively parallel to arms ED and EF of .

We need to show that

Let us extend BC to meet ED at point P.

We haveand. So, and are corresponding angles, these two should be equal.

Therefore,

Also, we have. So, and are consecutive interior angles, these two must be supplementary.

Therefore,

Hence proved.