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Page No 8.13:

Question 1:

In the given figure, OA and OB are opposite rays:


(i) If x = 25°, what is the value of y?

(ii) If y = 35°, what is the value of x?

Answer:

In figure:

Since OA and OB are opposite rays. Therefore, AB is a line. Since, OC stands on line AB.

Thus,and form a linear pair, therefore, their sum must be equal to.

Or, we can say that

From the given figure:

and

On substituting these two values, we get

                             ...(i)

(i) On puttingin (i), we get:

Hence, the value of y is.

(ii) On putting in in equation (A), we get:

Hence, the value of x is.

Page No 8.13:

Question 2:

In the given figure, write all pairs of adjacent angles and all the linear pairs.

Answer:

The figure is given as follows:

The following are the pair of adjacent angles:

and

and

The following are the linear pair:

and

and

Page No 8.13:

Question 3:

In the given figure, find x. Further find ∠BOC, ∠COD and∠AOD

Answer:

In the given figure:

AB is a straight line. Thus,, and form a linear pair.

Therefore their sum must be equal to.

We can say that

(i)

It is given that, and.

On substituting these values in (i), we get:

It is given that:

Therefore,

Also,

Therefore,

Therefore,

Page No 8.13:

Question 4:

In the given figure, rays OA, OB, OC, OD and OE have the common end point O. Show that ∠AOB + ∠BOC + ∠COD + ∠DOE + ∠EOA = 360°.

Answer:

Let us draw a straight line.

,and form a linear pair. Thus, their sum should be equal to.

Or, we can say that:

(I)

Similarly,,and form a linear pair. Thus, their sum should be equal to.

Or, we can say that:

(II)

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

Hence proved.

Page No 8.13:

Question 5:

In the given figure, ∠AOC and ∠BOC form a linear pair. If a − 2b = 30°, find a and b.

Answer:

In the figure given below, it is given thatand forms a linear pair.

Thus, the sum of and should be equal to.

Or, we can say that:

From the figure above, and

Therefore,

It is given that:

On comparing (i) and (ii), we get:

Putting in (i), we get :

Hence, the values for a and b areand respectively.

Page No 8.13:

Question 6:

How many pairs of adjacent angle are formed when two lines intersect in a point?

Answer:

Suppose we have two lines, say AB and CD intersect at a point, O as shown in the figure below:

 

Then there are 4 pairs of adjacent angles formed, namely:

  1. and

  2. and

  3. and

  4. and

Page No 8.13:

Question 7:

How many pairs of adjacent angles, in all, can you name in the given figure.

Answer:

In the given figure,

We have 10 adjacent angle pairs, namely:

  1. and

  2. and

  3. and

  4. and

  5. and

  6. and

  7. and

  8. and

  9. and

  10. and

Page No 8.13:

Question 8:

In the given figure, determine the value of x.

Answer:

In the given figure:

is a straight line. Thus,and form a linear pair.

Therefore their sum must be equal to.

We can say that

It is given that, substituting this value in equation above, we get:

Page No 8.13:

Question 9:

In the given figure, AOC is a line, find x.

Answer:

It is given that AOC is a line. Therefore, and form a linear pair. Thus, the sum of and must be equal to .

Or, we can say that

Also, and. On putting these values in the equation above we have:

Hence, the required value of is.



Page No 8.14:

Question 10:

In the given figure, POS is a line, find x.

Answer:

The figure is given as follows:

It is given that POS is a line.

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

It is given that, and. Therefore, we get:

600+4x+400=1800       4x+1000=1800                 4x=1800-1000                 4x=800                   x=8004                       x=200
Hence, the required value of x is.

Page No 8.14:

Question 11:

In Fig. 8.40, ACB is a line such that ∠DCA = 5x and ∠DCB = 4x.  Find the value of x.

Answer:

It is given that ACB is a line in the figure given below.

Thus,and form a linear pair.

Therefore, their sum must be equal to.

Or, we can say that

Also, and. This further simplifies to :

Hence, the value of x is 20o.

Page No 8.14:

Question 12:

In the given figure, ∠POR = 3x and ∠QOR = 2x + 10, find the value of x for which POQ will be a line.

Answer:

Here we have POQ as a line

So, andform a linear pair.

Therefore, their sum must be equal to.

Or, we can say that

It is given that and .On substituting these values above, we get :

Hence, the value of x is .

Page No 8.14:

Question 13:

In the given figure, a is greater than b by one third of a right-angle. Find the values of a and b.

Answer:

It is given that in the figure given below; a is greater than b by one-third of a right angle.

Or we can say that, the difference between a and b is.

That is;

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

We can say that:

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

On putting, in (i):

Hence, the values are and.

Page No 8.14:

Question 14:

What value of y would make AOB a line in the given figure, if ∠AOC = 4y and ∠BOC = (6y + 30)

Answer:

Let us assume,as a straight line.

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

We can say that:

Also, and. This further simplifies to:

Hence, the value of makesas a line.

Page No 8.14:

Question 15:

If the given figure, ∠AOF and ∠FOG form a linear pair.
EOB = ∠FOC = 90° and ∠DOC = ∠FOG = ∠AOB = 30°


(i) Find the measure of ∠FOE, ∠COB and ∠DOE.
(ii) Name all the right angles.
(iii) Name three pairs of adjacent complementary angles.
(iv) Name three pairs of adjacent supplementary angles.
(v) Name three pairs of adjacent angles.

Answer:

The given figure is as follows:

 

(i)

It is given that,,and form a linear pair .

Therefore, their sum must be equal to .

That is ,

It is given that :

,

and

in equation above, we get:

It is given that:

From the above figure:

Similarly, we have:

From the above figure:

(ii)

We have:

From the figure above and the measurements of the calculated angles we get two right angles as and.

Two right angles are already given asand.

(iii)

We have to find the three pair of adjacent complementary angles.

We know that is a right angle.

Therefore,

and are complementary angles.

Similarly, is a right angle.

Therefore,

and are complementary angles.

Similarly, is a right angle.

Therefore,

and are complementary angles.

(iv)

We have to find the three pair of adjacent supplementary angles.

Since,is a straight line.

Therefore, following are the three linear pair, which are supplementary:

and ;

and and

and

(v)

We have to find three pair of adjacent angles, which are as follows:

and

and

and



Page No 8.15:

Question 16:

In the given figure, OP, OQ, OR and OS are four rays. Prove that:
POQ + ∠QOR + ∠SOR + ∠POS = 360°

Answer:

Let us draw as a straight line.

 

Since,is a line, therefore,, and form a linear pair.

Also, and form a linear pair.

Thus, we have:

(i)

And

(ii)

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

Hence proved.

Page No 8.15:

Question 17:

In the given figure, ray OS stand on a line POQ, Ray OR and ray OT are angle bisectors of ∠POS and∠SOQ respectively. If ∠POS = x, find ∠ROT.

Answer:

In the figure given below, we have

Rayas the bisector of

Therefore,

Or,

(I)

 

Similarly, Rayas the bisector of

Therefore,

Or,

(II)

Also, Raystand on a line. Therefore,and form a linear pair.

Thus,

From (I) and (II):

Hence, the value of is 90°.

Page No 8.15:

Question 18:

In the given figure, lines PQ and RS intersect each other at point O. If ∠POR: ∠ROQ = 5 : 7, find all the angles.

Answer:

Let andbe and respectively.

Since, Ray stand on line.Thus, and form a linear pair.

Therefore, their sum must be equal to.

Or,

Thus,

Thus,

It is evident from the figure, thatand are vertically opposite angles.

And we know that vertically opposite angles are equal.

Therefore,

Similarly,and are vertically opposite angles.

And we know that vertically opposite angles are equal.

Therefore,

Page No 8.15:

Question 19:

In the given figure, POQ is a line. Ray OR is perpendicular to line PQ. OS is another ray lying between rays OP and OR. Prove that ∠ROS = 12 (∠QOSPOS).

Answer:

The given figure is as follows:

We have POQ as a line. Ray OR is perpendicular to line PQ. Therefore,

From the figure above, we get:

 (i)

and  form a linear pair. Therefore,

 (ii)

From (i) and (ii) equation we get:
QOS+POS=2×90

Hence proved.



Page No 8.19:

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:

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

Page No 8.19:

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:

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:



Page No 8.20:

Question 3:

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

Answer:

In the given question, the values of x, y, and z will be determined as follows:
z and 25° form a linear pair.
So, z + 25°=180°z=180-25 z=155°

Now, z and x are vertically opposite to each other. So, x155°.

Also, y and x form a linear pair. 
So, y+ 155°=180°y=180-155 y=25°

Hence, the values are x=155°, y=25° and z=155°.

Page No 8.20:

Question 4:

In the given figure, find the value of x.

Answer:

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, .

Page No 8.20:

Question 5:

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

Answer:

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.

Page No 8.20:

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.

Answer:

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.

Page No 8.20:

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.

Answer:

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.

Page No 8.20:

Question 8:

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


(i) Determine y when x = 60°

(ii) Determine x when y = 40

Answer:

Raysand intersect at point.

Therefore, and form a linear pair.

Thus,

(i)

On substituting:

(ii)

On substituting:

Page No 8.20:

Question 9:

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

Answer:

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):

Page No 8.20:

Question 10:

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:

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,

Page No 8.20:

Question 11:

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

Answer:

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:

Page No 8.20:

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, 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:

(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.



Page No 8.21:

Question 13:

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:

(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:



Page No 8.38:

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:

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
1=36×3=108° and 2=36×2=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.

Page No 8.38:

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:

According to the given figure,
m || n and are cut by transversal p.  
2=120°         (alternate interior angles are equal)
Also, l || m. So, 1=3         (corresponding angles)
Also, 3 and 120° form a linear pair.
3+120°=180°3=180-1203=60°

And 1=3=60°,2=120°     
 

Page No 8.38:

Question 3:

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

Answer:

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.



Page No 8.39:

Question 4:

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

Answer:

The figure is given as follows:

 

We need to prove that.

It is given that and

ACD=ACE+ECDACD=22°+35°ACD=57°

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 No 8.39:

Question 5:

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

Answer:

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.

Page No 8.39:

Question 6:

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

Answer:

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.

Page No 8.39:

Question 7:

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

Answer:

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,

Page No 8.39:

Question 8:

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

Answer:

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 .

Page No 8.39:

Question 9:

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

Answer:

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, .

Page No 8.39:

Question 10:

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:

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.

Page No 8.39:

Question 11:

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:

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.



Page No 8.40:

Question 12:

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

Answer:

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 .

Page No 8.40:

Question 13:

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

Answer:

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 .

Page No 8.40:

Question 14:

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

Answer:

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.

Page No 8.40:

Question 15:

In the given figure, ∠1 = 60° and ∠2 = 23rd of a right angle. Prove that l || m.

Answer:

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.

Page No 8.40:

Question 16:

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

Answer:

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 .

Page No 8.40:

Question 17:

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

Answer:

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 .

Page No 8.40:

Question 18:

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

Answer:

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:

Page No 8.40:

Question 19:

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

Answer:

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:

Page No 8.40:

Question 20:

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

Answer:

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.

Page No 8.40:

Question 21:

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

Answer:

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.

Page No 8.40:

Question 22:

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

Answer:

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 .



Page No 8.41:

Question 23:

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

Answer:

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.

Page No 8.41:

Question 24:

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:

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.

Page No 8.41:

Question 25:

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:

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.

Page No 8.41:

Question 26:

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:

(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.

Page No 8.41:

Question 27:

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:

(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.



Page No 8.42:

Question 1:

Define complementary angles.

Answer:

Complementary Angles: Two angles, the sum of whose measures is, are called complementary angles.

Thus, anglesand are complementary angles, if

Example 1:

Angles of measure andare complementary angles, because

Example 2:

Angles of measure andare complementary angles, because

Page No 8.42:

Question 2:

Define supplementary angles.

Answer:

Supplementary Angles: Two angles, the sum of whose measures is , are called supplementary angles.

Thus, anglesand are supplementary angles, if

 

Example 1:

Angles of measure andare supplementary angles, because

Example 2:

Angles of measure andare supplementary angles, because

Page No 8.42:

Question 3:

Define adjacent angles.

Answer:

Adjacent angles: Two angles are called adjacent angles, if:

  1. They have the same vertex,

  2. They have a common arm, and

  3. Uncommon arms are on either side of the common arm.

In the figure above,and have a common vertex.

Also, they have a common arm and the distinct arms and, lies on the opposite sides of the line.

Therefore, and are adjacent angles.

Page No 8.42:

Question 4:

The complement of an acute angle is ..............

Answer:

The complement of an acute angle is an acute angle.

Explanation:

As the sum of the complementary angles is.

Let one of the angle measures.

Then, other angle becomes, which is clearly an acute angle.

Page No 8.42:

Question 5:

The supplement of an acute angle is .................

Answer:

The supplement of an acute angle is an obtuse angle.

Explanation:

As the sum of the supplementary angles is.

Let one of the angle measures, such that

Let the other angle measures

As the angles are supplementary there sum is.

Then, other angle y is clearly an obtuse angle.

Illustration:

Let the given acute angle be

Then, the other angle becomes

This is clearly an obtuse angle.

Page No 8.42:

Question 6:

The supplement of a right angle is ..............

Answer:

We have to find the supplement of a right angle.

We know that a right angle is equal to.

Let the required angle be.

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

Thus, the require angle becomes

Page No 8.42:

Question 7:

Write the complement of an angle of measure x°.

Answer:

We have to write the complement of an angle which measures.

Let the other angle be.

We know that the sum of the complementary angles be 90°.

Therefore,

Page No 8.42:

Question 8:

Write the supplement of an angle of measure 2y°.

Answer:

Let the required angle measures

It is given that two angles measuring andare supplementary. Therefore, their sum must be equal to.

Or, we can say that:

Hence, the required angle measures.

Page No 8.42:

Question 9:

If a wheel has six spokes equally spaced, then find the measure of the angle between two adjacent spokes.

Answer:

It is given that the six spokes are equally spaced, thus, two adjacent spokes subtend equal angle at the centre of the wheel.

Let that angle measures

Also, the six spokes form a complete angle, that is,

Therefore,

Hence, the measure of the angle between two adjacent spokes measures.



Page No 8.43:

Question 10:

An angle is equal to its supplement. Determine its measure.

Answer:

Let the supplement of the angle be

According the given statement, the required angle is equal to its supplement, therefore, the required angle becomes.

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

Or we can say that:

Hence, the required angle measures .

Page No 8.43:

Question 11:

An angle is equal to five times its complement. Determine its measure.

Answer:

Let the complement of the required angle measures

Therefore, the required angle becomes

Since, the angles are complementary, thus, their sum must be equal to.

Or we can say that :

Hence, the required angle becomes:

Page No 8.43:

Question 12:

How many pairs of adjacent angles are formed when two lines intersect in a point?

Answer:

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

We have the following pair of adjacent angles, so formed:

and

and

and

and

Hence, in total four pair of adjacent angles are formed.

Page No 8.43:

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:

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).

Page No 8.43:

Question 2:

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:

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) .

Page No 8.43:

Question 3:

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:

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

Thus, AOD,AOC,COB and BOD form a complete angle, that is the sum of these four angle is.

That is,

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

It is given that

                               ...(ii)

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

Hence, the correct choice is (a).

Page No 8.43:

Question 4:

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:

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).

Page No 8.43:

Question 5:

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:

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).

Page No 8.43:

Question 6:

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:

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).

Page No 8.43:

Question 7:

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

(a) 85°

(b) 90°

(c) 95°

(d) 100°

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

AOC+BOD=85°                      ...(ii)

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

COD+85°=180°COD=180°-85°COD=95°
Hence, (c) is the correct option.

 



Page No 8.44:

Question 8:

In the given figure, the value of y is

(a) 20°

(b) 30°

(c) 45°

(d) 60°

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).

Page No 8.44:

Question 9:

In the given figure, if yx = 5 and zx = 4, then the value of x is

(a) 8°

(b) 18°

(c) 12°

(d) 15°

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).

Page No 8.44:

Question 10:

In the given figure, the value of x is

(a) 12

(b) 15

(c) 20

(d) 30

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).

Page No 8.44:

Question 11:

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:

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).



Page No 8.45:

Question 12:

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:

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).

Page No 8.45:

Question 13:

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

(a) 85°

(b) 135°

(c) 145°

(d) 215°

Answer:

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).

Page No 8.45:

Question 14:

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

(a) 20°

(b) 30°

(c) 45°

(d) 60°

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).



Page No 8.46:

Question 15:

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:

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).

Page No 8.46:

Question 16:

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

(a) 70°

(b) 80°

(c) 90°

(d) 180°

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).

Page No 8.46:

Question 17:

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:

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).

Page No 8.46:

Question 18:

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

Answer:

In the given figure:

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

That is;

Also, 1 and 2 are vertically opposite angles, therefore,

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

Hence, the correct choice is (b) .

Page No 8.46:

Question 19:

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



(a) 180 − w + z

(b) 180 + wz

(c) 180 − w z

(d) 180 + w + z

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).



Page No 8.47:

Question 20:

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



(a) 100

(b) 120

(c) 135

(d) 150

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).

Page No 8.47:

Question 21:

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-x2

(d) 90 − 2x


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).

Page No 8.47:

Question 22:

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



(a) 60

(b) 50

(c) 45

(d) 30

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).



Page No 8.48:

Question 23:

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:

The figure is given as follows:

It is given that AB is parallel to CD.

Thus,and BDC are consecutive interior angles.

Therefore, their sum must be supplementary.

That is,

 ABD+BDC=180°

From the figure, we get:

Hence, the correct choice is (d).

Page No 8.48:

Question 24:

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

(a) 130°

(b) 105°

(c) 175°

(d) 125°

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).

Page No 8.48:

Question 25:

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:

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).

Page No 8.48:

Question 26:

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

(a) 20°

(b) 45°

(c) 65°

(d) 85°

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).



Page No 8.49:

Question 27:

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

(a) 100°

(b) 105°

(c) 110°

(d) 115°

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).

Page No 8.49:

Question 28:

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

(a) 70°

(b) 100°

(c) 40°

(d) 30°

Answer:

We have the following figure:

It is given that

We know that consecutive interior angles are supplementary.

Therefore,


1=AOB=110   (vertically opposite angles)

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

In ΔAOB:

30°+x+110°=180°x=180-110-30=40

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

Page No 8.49:

Question 29:

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

(a) 105°

(b) 65°

(c) 40°

(d) 25°

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).

Page No 8.49:

Question 30:

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:

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 No 8.7:

Question 1:

Write the complement of each of the following angles:

(i) 20°

(ii) 35°

(iii) 90°

(iv) 77°

(v) 30°

Answer:

(i) Let the complement of angle measures x°

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

Or we can say that

Hence, the complement of angle measures

(ii) Let the complement of angle measures x°

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

Or we can say that

Hence, the complement of angle measures

(iii) Let the complement of angle measures x°

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

Or we can say that

Hence, the complement of angle measures

(iv) Let the complement of angle measures x°

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

Or we can say that

Hence, the complement of angle measures

(v) Let the complement of angle measures x°

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

Or we can say that

Hence, the complement of angle measures.

Page No 8.7:

Question 2:

Write the supplement of each of the following angles:

(i) 54°

(ii) 132°

(iii) 138°

Answer:

(i) Let the supplement of angle measures x°

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

Or we can say that

Hence, the supplement of angle measures.

(ii) Let the supplement of angle measures x°

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

Or we can say that

Hence, the supplement of angle measures.

(iii) Let the supplement of angle measures x°

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

Or we can say that

Hence, the supplement of angle measures.

Page No 8.7:

Question 3:

If an angle is 28° less than its complement, find its measure.

Answer:

Let one angle be x°.

Then the required angle becomes

It is given that x° andare complementary

Therefore their sum must be equal to

On dividing both sides of the equation by 2,we get:

Also

Hence the measure of the required angle is.

Page No 8.7:

Question 4:

If an angle is 30° more than one half of its complement, find the measure of the angle.

Answer:

Let the measure of the required angle be x°.

Thus its complement becomes

According to the statement, the required angle is 30 more than half of its complementary angle that is; the required angle x becomes,

.

Thus

Taking 2 on left hand side of the equation, we get

Hence, the required angle measures.

Page No 8.7:

Question 5:

Two supplementary angles are in the ratio 4:5. Find the angles.

Answer:

Let the two angles be 4x and 5x.

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

This can also be written as

Dividing both sides of equation by 9, we get

The two angles become

Also,

Hence,and are the measure of two supplementary angles.

Page No 8.7:

Question 6:

Two supplementary angles differ by 48°. Find the angles.

Answer:

Let one angle measures. Then, the second angle becomes.

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

Thus,

On dividing both sides of the equation by, we get

Also,

Hence, the required angles measureand.

Page No 8.7:

Question 7:

An angle is equal to 8 times its complement. Determine its measure.

Answer:

Let the required angle be x°

Thus its complement becomes

It is given that the angle x is 8 times its complementary angle, this means

Hence, the required angle measures.

Page No 8.7:

Question 8:

If the angles (2x − 10)° and (x − 5)° are complementary angles, find x.

Answer:

It is given that and are complementary angles.

Therefore, their sum must be equal to 90°.

Thus,

Hence the value of x is.

Page No 8.7:

Question 9:

If the complement of an angle is equal to the supplement of the thrice of it. Find the measure of the angle.

Answer:

Let the angle measures x°

Therefore, the measure of its complementary angle becomes

Also, supplement of its thrice means

According to the question,

Hence, the required angle measures.

Page No 8.7:

Question 10:

If an angle differs from its complement by 10°, find the angle.

Answer:

Let the angle measures x°

Therefore, the measure of its complement becomes

According to the question the above mentioned complementary angles differ by 10°.

Thus,

Hence the required angle measures.

Page No 8.7:

Question 11:

If the supplement of an angle is three times its complement, find the angle.

Answer:

Let the angle measures x°

Therefore, the measure of its complement isand measure of its supplement is

According to the question the supplement of is three times the complement, this means

Hence, the required angle measures.

Page No 8.7:

Question 12:

If the supplement of an angle is two-third of itself. Determine the angle and its supplement.

Answer:

Let the angle measures x°.

Therefore, the measure of its supplement is

It is given that the supplement is two third of itself, this means

Now, let’s calculate the supplement

Hence, the measure of the angle and its supplement areandrespectively.

Page No 8.7:

Question 13:

An angle is 14° more than its complementary angle. What is its measure?

Answer:

Let the angle measures x°

Therefore, the measure of its complement becomes

According to the given statement, the angle is 14 more than its complement.

Thus we have,

The measure of its complement becomes

Hence, the required angle measures and its complement measures.

Page No 8.7:

Question 14:

The measure of an angle is twice the measure of its supplementary angle. Find its measure.

Answer:

Let the angle measures x°

Therefore, the measure of its supplement becomes

According to the given statement, the required angle is twice the supplement.

Thus

Hence the required angle measures.



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