Rd Sharma 2020 2021 Solutions for Class 9 Maths Chapter 10 Lines And Angles are provided here with simple stepbystep explanations. These solutions for Lines And Angles are extremely popular among Class 9 students for Maths Lines And Angles Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rd Sharma 2020 2021 Book of Class 9 Maths Chapter 10 are provided here for you for free. You will also love the adfree experience on Meritnation’s Rd Sharma 2020 2021 Solutions. All Rd Sharma 2020 2021 Solutions for class Class 9 Maths are prepared by experts and are 100% accurate.
Page No 10.14:
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 10.14:
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 10.14:
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 10.14:
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 10.15:
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 10.15:
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:

and

and

and

and
Page No 10.15:
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:

and

and

and

and

and

and

and

and

and

and
Page No 10.15:
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 10.15:
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 10.15:
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:
${60}^{0}+4x+{40}^{0}={180}^{0}\phantom{\rule{0ex}{0ex}}4x+{100}^{0}={180}^{0}\phantom{\rule{0ex}{0ex}}4x={180}^{0}{100}^{0}\phantom{\rule{0ex}{0ex}}4x={80}^{0}\phantom{\rule{0ex}{0ex}}x=\frac{{80}^{0}}{4}\phantom{\rule{0ex}{0ex}}x=\overline{){20}^{0}}\phantom{\rule{0ex}{0ex}}$
Hence, the required value of x is.
Page No 10.15:
Question 11:
In the given figure, ACB is a line such that ∠DCA = 5x and ∠DCB = 4x. Find the values of ∠DCA and ∠DCB.
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 :
$\therefore \angle DCA=5x=5\times 20\xb0=100\xb0\phantom{\rule{0ex}{0ex}}\angle DCB=4x=4\times 20\xb0=80\xb0$
Hence, the values of ∠DCA and ∠DCB are 100^{∘} and 80^{∘} respectively.
Page No 10.15:
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 10.15:
Question 13:
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 10.16:
Question 14:
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 10.16:
Question 15:
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 10.16:
Question 16:
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 10.16:
Question 17:
In the given figure, a is greater than b by one third of a rightangle. Find the values of a and b.
Answer:
It is given that in the figure given below; a is greater than b by onethird 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 10.16:
Question 18:
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 10.17:
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 = $\frac{1}{2}$ (∠QOS − POS).
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:
$\angle QOS+\angle POS=2\times 90$
Hence proved.
Page No 10.22:
Question 1:
In the given figure, lines l_{1} and l_{2} 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 10.22:
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 10.22:
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 $25\xb0$ form a linear pair.
$\mathrm{So},z+25\xb0=180\xb0\phantom{\rule{0ex}{0ex}}\Rightarrow z=18025\phantom{\rule{0ex}{0ex}}\Rightarrow z=155\xb0\phantom{\rule{0ex}{0ex}}$
Now, z and x are vertically opposite to each other. So, x = $155\xb0$.
Also, y and x form a linear pair.
$\mathrm{So},y+155\xb0=180\xb0\phantom{\rule{0ex}{0ex}}\Rightarrow y=180155\phantom{\rule{0ex}{0ex}}\Rightarrow y=25\xb0\phantom{\rule{0ex}{0ex}}$
Hence, the values are $x=155\xb0,y=25\xb0\mathrm{and}z=155\xb0$.
Page No 10.22:
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 10.22:
Question 5:
If one of the four angles formed by two intersecting lines is a right angle, then show that each of the four angles is a right angle.
Answer:
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 10.23:
Question 6:
In the given figure, rays AB and CD intersect at O.
(i) Determine y when x = 60°
(ii) Determine x when y = 40
Answer:
Raysand intersect at point.
Therefore, and form a linear pair.
Thus,
(i)
On substituting:
(ii)
On substituting:
Page No 10.23:
Question 7:
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 10.23:
Question 8:
AB, CD and EF are three concurrent lines passing through the point O such that OF bisects ∠BOD. If ∠BOF = 35°, find ∠BOC and ∠AOD.
Answer:
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 10.23:
Question 9:
In the given figure, lines AB and CD intersect at O. If ∠AOC + ∠BOE = 70° and ∠BOD = 40°, find ∠BOE and reflex ∠COE.
Answer:
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 10.23:
Question 10:
Which of the following statements are true (T) and which are false (F)?
(i) Angles forming a linear pair are supplementary.
(ii) If two adjacent angles are equal, then each angle measures 90°.
(iii) Angles forming a linear pair can both be acute angles.
(iv) If angles forming a linear pair are equal, then each of these angles is of measure 90°.
Answer:
(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 10.23:
Question 11:
Fill in the blanks so as to make the following statements true:
(i) If one angle of a linear pair is acute, then its other angle will be ........
(ii) A ray stands on a line, then the sum of the two adjacent angles so formed is ..........
(iii) If the sum of two adjacent angles is 180°, then the ........ arms of the two angles are opposite rays.
Answer:
(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 10.23:
Question 12:
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 10.23:
Question 13:
If two straight lines intersect each other, prove that the ray opposite to the bisector of one of the angles thus formed bisects the vertically opposite angle.
Answer:
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 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:
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
$\angle 1=36\times 3=108\xb0\mathrm{and}\angle 2=36\times 2=72\xb0$
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 10.46:
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.
$\angle 2=120\xb0(\mathrm{alternate}\mathrm{interior}\mathrm{angles}\mathrm{are}\mathrm{equal})$
Also, l  m. So, $\angle 1=\angle 3(\mathrm{corresponding}\mathrm{angles})$
Also, $\angle 3\mathrm{and}120\xb0\mathrm{form}\mathrm{a}\mathrm{linear}\mathrm{pair}.$
$\angle 3+120\xb0=180\xb0\phantom{\rule{0ex}{0ex}}\Rightarrow \angle 3=180120\phantom{\rule{0ex}{0ex}}\Rightarrow \angle 3=60\xb0$
And $\angle 1=\angle 3=60\xb0,\angle 2=120\xb0$
Page No 10.46:
Question 3:
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 10.46:
Question 4:
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 10.47:
Question 5:
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 10.47:
Question 6:
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 10.47:
Question 7:
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 10.47:
Question 8:
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 10.47:
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:
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 10.47:
Question 10:
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 10.47:
Question 11:
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 10.47:
Question 12:
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 10.47:
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:
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 10.47:
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:
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 10.47:
Question 15:
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 10.48:
Question 16:
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 10.48:
Question 17:
If l, m, n are three lines such that l  m and n $\perp $ l. prove that n $\perp $ 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 10.48:
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:
(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 10.48:
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:
(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 10.48:
Question 20:
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 10.48:
Question 21:
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
$\angle ACD=\angle ACE+\angle ECD\phantom{\rule{0ex}{0ex}}\angle ACD=22\xb0+35\xb0\phantom{\rule{0ex}{0ex}}\angle ACD=57\xb0$
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 10.49:
Question 22:
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 10.49:
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:
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 10.49:
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:
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 10.49:
Question 25:
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 10.49:
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:
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 10.49:
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:
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 10.50:
Question 1:
One angle is equal to three times its supplement. The measure of the angle is
(a) 130°
(b) 135°
(c) 90°
(d) 120°
Answer:
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 10.50:
Question 2:
Two straight line AB and CD intersect one another at the point O. If ∠AOC + ∠COB + ∠BOD = 274°, then ∠AOD =
(i) 86°
(ii) 90°
(iii) 94°
(iv) 137°
Answer:
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,
$\angle AOD+\angle AOC+\angle COB+\angle BOD=360\xb0$ ... (i)
It is given that
...(ii)
Subtracting (ii) from (i), we get:
Hence, the correct choice is (a).
Page No 10.50:
Question 3:
Two straight lines AB and CD cut each other at O. If ∠BOD = 63°, then ∠BOC =
(a) 63°
(b) 117°
(c) 17°
(d) 153°
Answer:
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 10.50:
Question 4:
Consider the following statements:
When two straight lines intersect:
(i) adjacent angles are complementary
(ii) adjacent angles are supplementary
(iii) opposite angles are equal
(iv) opposite angles are supplementary
Of these statements
(a) (i) and (ii) are correct
(b) (ii) and (iii) are correct
(c) (i) and (iv) are correct
(d) (ii) and (iv) are correct
Answer:
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 10.51:
Question 5:
Given ∠POR = 3x and ∠QOR = 2x + 10°. If POQ is a straight line, then the value of x is
(a) 30°
(b) 34°
(c) 36°
(d) none of these
Answer:
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 10.51:
Question 6:
In the given figure, AOB is a straight line. If ∠AOC + ∠BOD = 85°, then ∠COD =
(a) 85°
(b) 90°
(c) 95°
(d) 100°
Answer:
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
$\angle AOC+\angle BOD=85\xb0$ ...(ii)
On substituting the value of (ii) in (i) we get,
$\angle COD+85\xb0=180\xb0\phantom{\rule{0ex}{0ex}}\Rightarrow \angle COD=180\xb085\xb0\phantom{\rule{0ex}{0ex}}\Rightarrow \angle COD=95\xb0$
Hence, (c) is the correct option.
Page No 10.51:
Question 7:
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 10.51:
Question 8:
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 10.52:
Question 9:
In the given figure, which of the following statements must be true?
(i) a + b = d + c
(ii) a + c + e = 180°
(iii) b + f = c + e
(a) (i) only
(b) (ii) only
(c) (iii) only
(d) (ii) and (iii) only
Answer:
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 10.52:
Question 10:
If two interior angles on the same side of a transversal intersecting two parallel lines are in the ratio 2:3, then the measure of the larger angle is
(a) 54°
(b) 120°
(c) 108°
(d) 136°
Answer:
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 10.52:
Question 11:
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 10.52:
Question 12:
Two lines AB and CD intersect at O. If ∠AOC + ∠COB + ∠BOD = 270°, then ∠AOC =
(a) 70°
(b) 80°
(c) 90°
(d) 180°
Answer:
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 = $90\xb0$
Hence, the correct choice is (c).
Page No 10.52:
Question 13:
In the given figure, PQ  RS, ∠AEF = 95°, ∠BHS = 110° and ∠ABC = x°. Then the value of x is
(a) 15°
(b) 25°
(c) 70°
(d) 35°
Answer:
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 10.53:
Question 14:
In the given figure, if l_{1}  l_{2}, what is the value of x?
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) .
Page No 10.53:
Question 15:
In the given figure, if l_{1}  l_{2}, what is x + y in terms of w and z?
(a) 180 − w + z
(b) 180 + w − z
(c) 180 − w − z
(d) 180 + w + z
Answer:
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 10.53:
Question 16:
In the given figure, if l_{1}  l_{2}, 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 10.54:
Question 17:
In the given figure, if l_{1}  l_{2} and l_{3}  l_{4}, what is y in terms of x?
(a) 90 + x
(b) 90 + 2x
(c) $90\frac{x}{2}$
(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 10.54:
Question 18:
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 10.54:
Question 19:
In the given figure, if AB  HF and DE  FG, then the measure of ∠FDE is
(a) 108°
(b) 80°
(c) 100°
(d) 90°
Answer:
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 10.54:
Question 20:
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 10.55:
Question 21:
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 10.55:
Question 22:
In the given figure, if lines l and m are parallel lines, then x =
(a) 70°
(b) 100°
(c) 40°
(d) 30°
Answer:
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:
$30\xb0+x+110\xb0=180\xb0\phantom{\rule{0ex}{0ex}}\Rightarrow x=18011030=40$
Hence, the value of x will be $40\xb0$.
Thus, (c) is the correct answer.
Page No 10.55:
Question 23:
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 10.55:
Question 24:
In the given figure, if lines l and m are parallel, then the value of x is
(a) 35°
(b) 55°
(c) 65°
(d) 75°
Answer:
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 10.56:
Question 25:
Two complementary angles are such that two times the measure of one is equal to three times the measure of the other. The measure of the smaller angle is
(a) 45°
(b) 30°
(c) 36°
(d) none of these
Answer:
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 10.56:
Question 26:
In the given figure, if $\frac{y}{x}=5$and $\frac{z}{x}=4$, then the value of x is
(a) 8°
(b) 18°
(c) 12°
(d) 15°
Answer:
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 10.56:
Question 27:
In the given figure, AB  CD  EF and GH  KL. The measure of ∠HKL is
(a) 85°
(b) 135°
(c) 145°
(d) 215°
Answer:
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 10.56:
Question 28:
AB and CD are two parallel lines. PQ cuts AB and CD at E and F respectively. EL is the bisector of ∠FEB. If ∠LEB = 35°, then ∠CFQ will be
(a) 55°
(b) 70°
(c) 110°
(d) 130°
Answer:
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 10.56:
Question 29:
In the given figure, If line segment AB is parallel to the line segment CD, what is the value of y?
(a) 12
(b) 15
(c) 18
(d) 20
Answer:
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,
$\angle ABD+\angle BDC=180\xb0$
From the figure, we get:
Hence, the correct choice is (d).
Page No 10.57:
Question 30:
In the given figure, if CP  DQ, then the measure of x is
(a) 130°
(b) 105°
(c) 175°
(d) 125°
Answer:
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 10.57:
Question 1:
An angle is $\frac{2}{3}$ times its supplementary angle, the measure of the angle is __________.
Answer:
Let the measure of angle be x.
∴ Supplement of the angle = 180º − x
It is given that,
Measure of the angle = $\frac{2}{3}$ × Supplement of the angle
$\therefore x=\frac{2}{3}\left(180\xb0x\right)\phantom{\rule{0ex}{0ex}}\Rightarrow 3x=360\xb02x\phantom{\rule{0ex}{0ex}}\Rightarrow 5x=360\xb0\phantom{\rule{0ex}{0ex}}\Rightarrow x=\frac{360\xb0}{5}=72\xb0$
Thus, the measure of the angle is 72º.
An angle is $\frac{2}{3}$ times its supplementary angle, the measure of the angle is ___72º___.
Page No 10.57:
Question 2:
If two complementary angles are in the ratio 7 : 11, the supplement of the larger angle is _________.
Answer:
Let the two complementary angle be 7x and 11x.
Now,
7x + 11x = 90º (Two angles are complementary if their sum is 90º)
⇒ 18x = 90º
$\Rightarrow x=\frac{90\xb0}{18}=5\xb0$
$\therefore 7x=7\times 5\xb0=35\xb0$ and $11x=11\times 5\xb0=55\xb0$
So, the measure of the larger angle is 55º.
∴ Supplement of the larger angle = 180º − 55º = 125º
If two complementary angles are in the ratio 7 : 11, the supplement of the larger angle is ___125º___.
Page No 10.57:
Question 3:
If x° is the measure of an angle which is equal to its complement and y° is the measure of an angle which is equal to its supplement, then $\frac{x\xb0}{y\xb0}$ is equal to ___________.
Answer:
Two angles are complementary if their sum is 90º.
$\therefore x\xb0=90\xb0x\xb0\phantom{\rule{0ex}{0ex}}\Rightarrow 2x\xb0=90\xb0\phantom{\rule{0ex}{0ex}}\Rightarrow x\xb0=45\xb0$
Two angles are supplementary if their sum is 180º.
$\therefore y\xb0=180\xb0y\xb0\phantom{\rule{0ex}{0ex}}\Rightarrow 2y\xb0=180\xb0\phantom{\rule{0ex}{0ex}}\Rightarrow y\xb0=90\xb0$
$\therefore \frac{x\xb0}{y\xb0}=\frac{45\xb0}{90\xb0}=\frac{1}{2}$
If x° is the measure of an angle which is equal to its complement and y° is the measure of an angle which is equal to its supplement, then $\frac{x\xb0}{y\xb0}$ is equal to $\overline{)\frac{1}{2}}$.
Page No 10.57:
Question 4:
In the given figure, if ABCD  EF, PQRS, ∠ROD = 25° and ∠CQP = 60°, then ∠QRS = ____________.
Answer:
In the given figure, AB  CD and RQ is the transversal.
∴ ∠ARQ = ∠RQD (Pair of alternate interior angles)
∠RQD = 25º (Given)
⇒ ∠ARQ = 25º .....(1)
Also, PQ  RS and QR is the transversal.
∴ ∠BRS = ∠PQC (Pair of alternate exterior angles)
∠PQC = 60º (Given)
⇒ ∠BRS = 60º
Also,
∠BRS + ∠ARS = 180º (Linear pair)
⇒ 60º + ∠ARS = 180º
⇒ ∠ARS = 180º − 60º = 120º .....(2)
Now,
∠QRS = ∠ARQ + ∠ARS = 25º + 120º = 145º [Using (1) and (2)]
Thus, the measure of ∠QRS is 145º.
In the given figure, if AB CD  EF, PQ  RS, ∠ROD = 25° and ∠CQP = 60°, then ∠QRS = ___145º___.
Page No 10.58:
Question 5:
In the given figure, POQ is a line. The value of x is ___________.
Answer:
It is given that, POQ is a straight line.
∴ ∠POR + ∠ROQ = 180º (Linear pair)
⇒ 40º + (4x + 3x) = 180º (∠ROQ = ∠ROS + ∠SOQ)
⇒ 7x = 180º − 40º = 140º
⇒ x = $\frac{140\xb0}{7}$ = 20º
Thus, the value of x is 20º.
In the given figure, POQ is a line. The value of x is ___20º___.
Page No 10.58:
Question 6:
If two interior angles on the same side of a transversal intersecting two parallel lines are in the ratio 2 : 3, then the greater of the two angles is ________.
Answer:
Let the two interior angle be 2x and 3x.
If a transversal intersects two parallel lines, then each pair of consecutive interior angles are supplementary.
$\therefore 2x+3x=180\xb0\phantom{\rule{0ex}{0ex}}\Rightarrow 5x=180\xb0\phantom{\rule{0ex}{0ex}}\Rightarrow x=\frac{180\xb0}{5}=36\xb0$
So, 2x = 2 × 36º = 72º and 3x = 3 × 36º = 108º
Thus, the greater of the two angles is 108º.
If two interior angles on the same side of a transversal intersecting two parallel lines are in the ratio 2 : 3, then the greater of the two angles is ___108º___.
Page No 10.58:
Question 7:
The bisectors of two adjacent supplementary angels are always at ________.
Answer:
OP is the bisector of ∠BOC.
∴ ∠BOP = ∠COP = $\frac{\angle \mathrm{BOC}}{2}$ .....(1)
OQ is the bisector of ∠AOC
∴ ∠AOQ = ∠COQ = $\frac{\angle \mathrm{AOC}}{2}$ .....(2)
It is given that,
∠AOC and ∠BOC are two adjacent supplementary angles.
∴ ∠BOC + ∠AOC = 180º
⇒ 2∠POC + 2∠COQ = 180º
⇒ ∠POC + ∠COQ = $\frac{180\xb0}{2}$ = 90º
⇒ ∠POQ = 90º
Thus, the bisectors of two adjacent supplementary angles are always at right angle i.e. 90º.
The bisectors of two adjacent supplementary angels are always at __right angle__.
Page No 10.58:
Question 8:
The bisectors of a pair of vertically opposite angels are in the __________.
Answer:
Let AB and CD be two lines intersecting at O. Suppose OP and OQ are the bisectors of the vertically opposite angles ∠AOD and ∠BOC, respectively.
∴ ∠AOP = ∠DOP .....(1)
∠BOQ = ∠COQ .....(2)
Also, ∠BOD = ∠AOC .....(3) (Vertically opposite angles)
Now,
∠AOP + ∠DOP + ∠BOD + ∠BOQ + ∠COQ + ∠AOC = 360º (Sum of angles formed at a point is 360º)
⇒ ∠DOP + ∠DOP + ∠BOD + ∠BOQ + ∠BOQ + ∠BOD = 360º [Using (1), (2) and (3)]
⇒ 2∠DOP + 2∠BOD + 2∠BOQ = 360º
⇒ 2(∠DOP + ∠BOD + ∠BOQ) = 360º
⇒ ∠POQ = $\frac{360\xb0}{2}$ = 180º
Thus, OP and OQ i.e. the bisectors of the pair of vertically opposite angles are in the same line.
The bisectors of a pair of vertically opposite angels are in the __same line__.
Page No 10.58:
Question 9:
If a transversal intersects two parallel lines then each pair of consecutive interior angle is _________.
Answer:
AB and CD are two parallel lines intersected by a transversal PQ at R and S, respectively. Here, the pair of consecutive interior angles are ∠BRS and ∠DSR & ∠ARS and ∠CSR.
Now,
∠PRB + ∠BRS = 180º .....(1) (Linear pair)
∠PRB = ∠DSR .....(2) (Corresponding angles)
From (1) and (2), we have
∠DSR + ∠BRS = 180º
Similarly, ∠ARS + ∠CSR = 180º
Thus, if a transversal intersects two parallel lines, then each pair of consecutive interior angles are supplementary.
If a transversal intersects two parallel lines then each pair of consecutive interior angle is __supplementary__.
Page No 10.58:
Question 10:
The angle whose supplement is three times the complement is __________.
Answer:
Let the measure of angle be x.
Supplement of the angle = 180º − x
Complement of the angle = 90º − x
It is given that,
Supplement of the angle = 3 × Complement of the angle
⇒ 180º − x = 3(90º − x)
⇒ 180º − x = 270º − 3x
⇒ 3x − x = 270º − 180º
⇒ 2x = 90º
⇒ x = $\frac{90\xb0}{2}$ = 45º
The angle whose supplement is three times the complement is ____45º____.
Page No 10.58:
Question 11:
The supplement of an angle and the complement of another have a sum equal to half of a complete angle. If the greater angle is 10° more than the smaller, then the measures of the angles are ___________.
Answer:
Let the two angles be x and y (x > y).
Supplement of angle x = 180º − x
Complement of angle y = 90º − y
Now,
180º − x + 90º − y = $\frac{1}{2}$ × 360º
⇒ 270º − x − y = 180º
⇒ x + y = 90º .....(1)
Also,
x = y + 10º
⇒ x − y = 10º .....(2)
Adding (1) and (2), we get
x + y + x − y = 90º + 10º
⇒ 2x = 100º
⇒ x = $\frac{100\xb0}{2}$ = 50º
Substituting the value of x in (1), we get
50º + y = 90º
⇒ y = 90º − 50º = 40º
Thus, the measure of the angles are 40º and 50º.
The supplement of an angle and the complement of another have a sum equal to half of a complete angle. If the greater angle is 10° more than the smaller, then the measures of the angles are ___40º and 50º___.
Page No 10.58:
Question 12:
If two parallel lines are intersected by a transversal then the bisectors of the interior angles form a _________.
Answer:
AB and CD are two parallel lines intersected by a transversal PQ at R and S, respectively.
RX is the bisector of ∠BRS and SX is the bisector of ∠DSR.
∴ ∠BRS = 2∠XRS .....(1)
∠DSR = 2∠XSR .....(2)
RY is the bisector of ∠ARS and SY is the bisector of ∠CSR.
∴ ∠ARS = 2∠YRS .....(3) (RY is the bisector of ∠ARS)
∠CSR = 2∠YSR .....(4) (SY is the bisector of ∠CSR)
We know that if a transversal intersects two parallel lines, then each pair of consecutive interior angles are supplementary.
∴ ∠BRS + ∠DSR = 180º
⇒ 2∠XRS + 2∠XSR = 180º [Using (1) and (2)]
⇒ 2(∠XRS + ∠XSR) = 180º
⇒ ∠XRS + ∠XSR = 90º .....(5)
In ∆RSX,
∠XRS + ∠XSR + ∠RXS = 180º (Angle sum property of triangle)
⇒ 90º + ∠RXS = 180º [Using (5)]
⇒ ∠RXS = 180º − 90º = 90º .....(6)
Similarly, ∠RYS = 90º .....(7)
Now,
∠BRS + ∠ARS = 180º (Linear pair)
⇒ 2∠XRS + 2∠YRS = 180º [Using (1) and (3)]
⇒ 2(∠XRS + ∠YRS) = 180º
⇒ ∠XRY = 90º .....(8)
Similarly, ∠XSY = 90º .....(9)
In quadrilateral XRYS,
∠RXS = ∠XRY = ∠RYS = ∠XSY = 90º [From (6), (7), (8) and (9)]
∴ Quadrilateral XRYS is a rectangle. (If each angle of a quadrilateral is 90º, then it is a rectangle)
If two parallel lines are intersected by a transversal then the bisectors of the interior angles form a __rectangle__.
Page No 10.58:
Question 13:
AB is a straight line and P is a point on it. If a line PQ is drawn not coinciding with PA and PB, then ∠APQ and ∠BPQ are ___________.
Answer:
Here, ∠APQ and ∠BPQ are adjacent angles. Also, AP and BP are two opposite rays.
We know that two adjacent angles are said to form a linear pair, if their noncommon arms are two opposite rays.
∴ ∠APQ + ∠BPQ = 180º (Linear pair of angles)
AB is a straight line and P is a point on it. If a line PQ is drawn not coinciding with PA and PB, then ∠APQ and ∠BPQ are __supplementary__.
Page No 10.58:
Question 14:
If the arms of one angle are respectively parallel to the arms of another angle, then the two angles are either _______ or _______.
Answer:
Let ∠ABC and ∠PQR be two angles such that AB  PQ and BC  QR.
Suppose PQ and BC intersect at O.
Figure 1  Figure 2 
Consider figure 1.
Here,
∠ABC = ∠POC .....(1) (Corresponding angle)
∠POC = ∠PQR .....(2) (Corresponding angle)
From (1) and (2), we have
∠ABC = ∠PQR
Consider figure 2.
∠ABC + ∠QOB = 180º .....(3) (If two lines are parallel, then the pair of adjacent interior angles are supplementary)
Also, ∠QOB = ∠PQR .....(4) (Alternate angles)
From (3) and (4), we have
∠ABC + ∠PQR = 180º
If the arms of one angle are respectively parallel to the arms of another angle, then the two angles are either __equal__ or __supplementary__.
Page No 10.58:
Question 15:
In given figure, if OP  RS, ∠OPQ = 110° and ∠QRS = 130°, then ∠PQR = _________.
Answer:
Extend OP to intersect QR at T.
Now, OU  RS and RT is the transversal.
∴ ∠PTR = ∠SRT (Pair of alternate angles)
⇒ ∠PTR = 130º (∠SRT = 130º)
Now,
∠PTR + ∠PTQ = 180º (Linear pair)
⇒ 130º + ∠PTQ = 180º
⇒ ∠PTQ = 180º − 130º = 50º
In ∆PQT,
∠OPQ = ∠PTQ + ∠PQT (The exterior angle of a triangle is equal to the sum of its opposite interior angles)
⇒ 110º = 50º + ∠PQT
⇒ ∠PQT = 110º − 50º = 60º
Or ∠PQR = 60º
Thus, the measure of ∠PQR is 60º.
In given figure, if OP  RS, ∠OPQ = 110° and ∠QRS = 130°, then ∠PQR = ___60º___.
Page No 10.59:
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 10.59:
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 10.59:
Question 3:
Define adjacent angles.
Answer:
Adjacent angles: Two angles are called adjacent angles, if:

They have the same vertex,

They have a common arm, and

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 10.59:
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 10.59:
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 10.59:
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 10.59:
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 10.59:
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 10.59:
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 10.59:
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 10.59:
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 10.59:
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 10.7:
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 10.7:
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 10.7:
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 10.7:
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 10.7:
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 10.7:
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 10.7:
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 10.7:
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 10.7:
Question 9:
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 10.7:
Question 10:
If the supplement of an angle is twothird 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 10.7:
Question 11:
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 10.7:
Question 12:
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.
Page No 10.7:
Question 13:
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 10.7:
Question 14:
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.
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