RD Sharma solution class 7 chapter 14 Lines and Angles Exercise 14.1

Exercise 14.1

Page-14.6

Question 1:

Write down each pair of adjacent angles shown in Fig.

Answer 1:

Adjacent angles are the angles that have a common vertex and a common arm.
Following are the adjacent angles in the given figure:

$$\begin{gathered} \angle DOC \mathrm{and} \angle BOC \\ \angle COB \mathrm{and} \angle BOA \end{gathered}$$

Question 2:

In Fig., name all the pairs of adjacent angles.

Answer 2:

In figure (i), the adjacent angles are:

$$\begin{gathered} \angle \mathrm{EBA} \mathrm{and}\angle \mathrm{ABC} \\ \angle \mathrm{ACB} \mathrm{and} \angle \mathrm{BCF} \\ \angle \mathrm{BAC} \mathrm{and} \angle \mathrm{CAD} \end{gathered}$$

In figure (ii), the adjacent angles are:

$\angle$BAD and $\angle$DAC
$\angle$BDA and $\angle$CDA

Question 3:

In figure, write down: (i) each linear pair (ii) each pair of vertically opposite angles.

Answer 3:

(i) Two adjacent angles are said to form a linear pair of angles if their non-common arms are two opposite rays.
$\angle$1 and $\angle$3
$\angle$1 and $\angle$2
$\angle$4 and $\angle$3
$\angle$4 and $\angle$2
$\angle$5 and $\angle$6
$\angle$5 and $\angle$7
$\angle$6 and $\angle$8
$\angle$7 and $\angle$8

(ii) Two angles formed by two intersecting lines having no common arms are called vertically opposite angles.
$\angle$1 and $\angle$4
$\angle$2 and $\angle$3
$\angle$5 and $\angle$8
$\angle$6 and $\angle$7

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Question 4:

Are the angles 1 and 2 given in Fig. adjacent angles?

Answer 4:

No, because they have no common vertex.

Question 5:

Find the complement of each of the following angles:
(i) 35°
(ii) 72°
(iii) 45°
(iv) 85°

Answer 5:

Two angles are called complementary angles if the sum of those angles is 90°.

Complementary angles of the following angles are:

$$\begin{gathered} \left(\mathrm{i}\right) 90°-35°=55° \\ \left(\mathrm{ii}\right) 90°-72°=18° \\ \left(\mathrm{iii}\right) 90°-45°=45° \\ \left(\mathrm{iv}\right) 90°-85°=5° \end{gathered}$$

Question 6:

Find the supplement of each of the following angles:
(i) 70°
(ii) 120°
(iii) 135°
(iv) 90°

Answer 6:

Two angles are called supplementary angles if the sum of those angles is 180°.
Supplementary angles of the following angles are:

(i) 180° − 70° = 110°
(ii) 180° − 120° = 60°
(iii) 180° − 135° = 45°
(iv) 180° − 90° = 90°

Question 7:

Identify the complementary and supplementary pairs of angles from the following pairs:
(i) 25°, 65°
(ii) 120°, 60°
(iii) 63°, 27°
(iv) 100°, 80°

Answer 7:

Since
$$\begin{gathered} \left(\mathrm{i}\right) 25°+65°=90° , \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{e} \mathrm{this} \mathrm{is} \mathrm{c}\mathrm{omplementary} \mathrm{pair} \mathrm{of} \mathrm{a}\mathrm{ngle}. \\ \left(\mathrm{i}\mathrm{i}\right) 120°+\; 60°=\; 180°, \mathrm{t}\mathrm{herefore} \mathrm{t}\mathrm{his} \mathrm{i}\mathrm{s} \mathrm{s}\mathrm{upplementary} pair \mathrm{o}\mathrm{f} \mathrm{a}\mathrm{ngle}. \\ \left(\mathrm{i}\mathrm{i}\mathrm{i}\right) 63°+27°=\; 90°, \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{e} \mathrm{this} \mathrm{is} \mathrm{c}\mathrm{omplementary} \mathrm{pair} \mathrm{of} \mathrm{a}\mathrm{ngle}. \\ \left(\mathrm{i}\mathrm{v}\right) 100°+\; 80°=\; 180° , \mathrm{t}\mathrm{herefore} \mathrm{t}\mathrm{his} \mathrm{i}\mathrm{s} \mathrm{s}\mathrm{upplementary} pair \mathrm{o}\mathrm{f} \mathrm{a}\mathrm{ngle}. \end{gathered}$$

Therefore, (i) and (iii) are the pairs of complementary angles and (ii) and (iv) are the pairs of supplementary angles.

Question 8:

Can two angles be supplementary, if both of them be
(i) obtuse?
(ii) right?
(iii) acute?

Answer 8:

(i) No, two obtuse angles cannot be supplementary.
(ii) Yes, two right angles can be supplementary. ($\because \angle 90°+\angle 90°=\angle 180°$)
(iii) No, two acute angles cannot be supplementary.

Question 9:

Name the four pairs of supplementary angles shown in Fig.

Answer 9:

Following are the supplementary angles:
$\angle$AOC and $\angle$COB
$\angle$BOC and $\angle$DOB
$\angle$BOD and $\angle$DOA
$\angle$AOC and $\angle$DOA

Question 10:

In Fig., A, B, C are collinear points and ∠DBA = ∠EBA.

(i) Name two linear pairs
(ii) Name two pairs of supplementary angles.

Answer 10:

(i) Linear pairs:
$\angle$ABD and $\angle$DBC
$\angle$ABE and $\angle$EBC

Because every linear pair forms supplementary angles, these angles are:
$\angle$ABD and $\angle$DBC
$\angle$ABE and $\angle$EBC

Question 11:

If two supplementary angles have equal measure, what is the measure of each angle?

Answer 11:

Let x and y be two supplementary angles that are equal.
$\angle x=\angle y$
According to the question,
$$\begin{gathered} \angle x+\angle y=180° \\ \Rightarrow \angle x+\angle x=180° \\ \Rightarrow 2\angle x=180° \\ \Rightarrow \angle x=\frac{180°}{2}=90° \\ \therefore \angle x=\angle y=90° \end{gathered}$$

Question 12:

If the complement of an angle is 28°, then find the supplement of the angle.

Answer 12:

Let x be the complement of the given angle $28°$.
$$\begin{gathered} \therefore \angle x+28°=90° \\ \Rightarrow \angle x=90°-28°=62° \end{gathered}$$
So, supplement of the angle = $180°-62°=118°$

Question 13:

In Fig. 19, name each linear pair and each pair of vertically opposite angles:

Answer 13:

Two adjacent angles are said to form a linear pair of angles if their non-common arms are two opposite rays.

$\angle$1 and $\angle$2
$\angle$2 and $\angle$3
$\angle$3 and $\angle$4
$\angle$1 and $\angle$4
$\angle$5 and $\angle$6
$\angle$6 and $\angle$7
$\angle$7 and $\angle$8
$\angle$8 and $\angle$5
$\angle$9 and $\angle$10
$\angle$10 and $\angle$11
$\angle$11 and $\angle$12
$\angle$12 and $\angle$9

Two angles formed by two intersecting lines having no common arms are called vertically opposite angles.
$\angle$1 and $\angle$3
$\angle$4 and $\angle$2
$\angle$5 and $\angle$7
$\angle$6 and $\angle$8
$\angle$9 and $\angle$11
$\angle$10 and $\angle$12

Question 14:

In Fig., OE is the bisector of ∠BOD. If ∠1 = 70°, find the magnitudes of ∠2, ∠3 and ∠4.

Answer 14:

Since OE is the bisector of $\angle$BOD,
$$\begin{gathered} \therefore \angle \mathrm{DOE}=\angle \mathrm{EOB} \\ \angle 2+\angle 1+\angle \mathrm{EOB}=180° \left(\mathrm{Linear} \mathrm{Pair}\right) \\ \angle 2+2\angle 1=180° \left(\angle 1=\angle \mathrm{EOB}\right) \\ \Rightarrow \angle 2=180°-2\angle 1=180°-2\times 70°=180°-140°=40° \end{gathered}$$
$$\begin{gathered} \angle 4=\angle 2=40° \left(\mathrm{Vertically} \mathrm{opposite} \mathrm{angles}\right) \\ \angle 3=\angle \mathrm{DOB}=\angle 1+\angle \mathrm{EOB}=70°+70°=140° \left[\angle 3=\angle \mathrm{DOB} \left(\mathrm{Vertically} \mathrm{opposite} \mathrm{angles}\right)\right] \end{gathered}$$

Question 15:

One of the angles forming a linear pair is a right angle. What can you say about its other angle?

Answer 15:

One angle of a linear pair is the right angle, i.e., 90°.
∴ The other angle = 180°​ - 90° = 90​°

Question 16:

One of the angles forming a linear pair is an obtuse angle. What kind of angle is the other?

Answer 16:

If one of the angles of a linear pair is obtuse, then the other angle should be acute; only then can their sum be 180°.

Question 17:

One of the angles forming a linear pair is an acute angle. What kind of angle is the other?

Answer 17:

In a linear pair, if one angle is acute, then the other angle should be obtuse. Only then their sum can be 180°.

Page-14.8

Question 18:

Can two acute angles form a linear pair?

Answer 18:

No, two acute angles cannot form a linear pair because their sum is always less than 180°.

Question 19:

If the supplement of an angle is 65°; then find its complement.

Answer 19:

Let x be the required angle.
Then, we have: 
x + 65° = 180°
$\Rightarrow$x = 180° - 65° = 115°

The complement of angle x cannot be determined.

Question 20:

Find the value of x in each of the following figures.

Answer 20:

(i)
Since $\angle \mathrm{BOA}+\angle \mathrm{BOC}=180°$(Linear pair)
$\therefore \angle x=180°-\angle \mathrm{BOA}=180°-60°=120°$

(ii)
$$\begin{gathered} \mathrm{Since} \angle \mathrm{QOP}+\angle \mathrm{QOR}=180° \left(\mathrm{Linear} \mathrm{pair}\right) \\ \therefore 2x+3x=180° \\ \Rightarrow 5x=180° \\ \Rightarrow x=\frac{180°}{5}=36° \end{gathered}$$

(iii)
$$\begin{gathered} \mathrm{Since} \angle \mathrm{LOP}+\angle \mathrm{PON}+\angle \mathrm{NOM}=180° \left(\mathrm{Linear} \mathrm{pair}\right) \\ \therefore \angle \mathrm{PON}=180°-\angle \mathrm{LOP}-\angle \mathrm{NOM} \\ \Rightarrow x=180°-35°-60° \\ \Rightarrow x=180°-95°=85° \end{gathered}$$

(iv)
$$\begin{gathered} \mathrm{Since} \angle \mathrm{COD}+\angle \mathrm{DOE}+\angle \mathrm{EOA}+\angle \mathrm{AOB}+\angle \mathrm{BOC}=360° \left(\mathrm{Sum} \mathrm{of} \mathrm{all} \mathrm{angles} \mathrm{at} \mathrm{a} \mathrm{point}\right) \\ \therefore 83°+92°+75°+47°+x=360° \\ \Rightarrow 297°+x=360° \\ \Rightarrow x=360°-297°=63° \end{gathered}$$

(v)
$$\begin{gathered} 2x°+x°+2x°+3x°=180° \\ \Rightarrow 8x=180 \\ \Rightarrow x=\frac{180}{8}=22.5° \end{gathered}$$

(vi)
$$\begin{gathered} 3x°=105° \\ \Rightarrow x=\frac{105}{3}=35° \end{gathered}$$

Question 21:

In Fig. 22, it being given that ∠1 = 65°, find all other angles.

Answer 21:

$\angle 1=\angle 3$          (Vertically opposite angles)
$\therefore \angle 3=65°$
Since $\angle 1+\angle 2=180°$       (Linear pair)
$\therefore \angle 2=180°-65°=115°$
$\angle 2=\angle 4$          (Vertically opposite angles)
$\therefore \angle 4=\angle 2=115°$ and $\angle 3=65°$

Page-14.9

Question 22:

In Fig., 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 22:

$\angle$AOC + $\angle$BOC = 180°                   (Linear pair)
$$\begin{gathered} \Rightarrow \left(2y+5\right)+3x=180° \\ \Rightarrow 3x+2y=175° \end{gathered}$$
(i) If x = 25°, then
$$\begin{gathered} 3\times 25°+2y=175° \\ \Rightarrow 75°+2y=175° \\ \Rightarrow 2y=175°-75°=100° \\ \Rightarrow y=\frac{100°}{2}=50° \end{gathered}$$
(ii) If y = 35°, then
$$\begin{gathered} 3x+2\times 35°=175° \\ \Rightarrow 3x+70°=175° \\ \Rightarrow 3x=175°-70°=105° \\ \Rightarrow x=\frac{105°}{3}=35° \end{gathered}$$

Question 23:

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

Answer 23:

Adjacent angles:

$$\begin{gathered} \angle DOA \mathrm{and} \angle DOC \\ \angle DOC \mathrm{and} \angle BOC \end{gathered}$$

$$\begin{gathered} \angle AOD \mathrm{and} \angle DOB \\ \angle BOC \mathrm{and} \angle AOC \end{gathered}$$

Linear pairs of angles:

$$\begin{gathered} \angle AOD \mathrm{and} \angle DOB \\ \angle BOC \mathrm{and} \angle AOC \end{gathered}$$

Question 24:

In Fig. 25, find ∠x. Further find ∠BOC, ∠COD and ∠AOD.

Answer 24:

$$\begin{gathered} \angle AOD+\angle DOC+\angle COB=180°(Linear pair) \\ (x+10)°+x°+(x+20)°=180° \\ 3x+30°=180° \\ 3x=180°-30° \\ 3x=150° \\ x=\frac{150°}{3}=50° \end{gathered}$$
$$\begin{gathered} \angle BOC=x+20°=50°+20°=70° \\ \angle COD=x=50° \\ \angle AOD=x+10°=50°+10°=60° \end{gathered}$$

Question 25:

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

Answer 25:

If two lines intersect at a point, then four adjacent pairs are formed, and those pairs are linear as well.

Question 26:

How many pairs of adjacent angles, in all, can you name in Fig.?

Answer 26:

There are 10 adjacent pairs in the given figure; they are:
$$\begin{gathered} \angle \mathrm{EOD} \mathrm{and} \angle \mathrm{DOC} \\ \angle \mathrm{COD} \mathrm{and} \angle \mathrm{BOC} \\ \angle \mathrm{COB} \mathrm{and} \angle \mathrm{BOA} \end{gathered}$$
$$\begin{gathered} \angle \mathrm{AOB} \mathrm{and} \angle \mathrm{BOD} \\ \angle \mathrm{BOC} \mathrm{and} \angle \mathrm{COE} \\ \angle \mathrm{COD} \mathrm{and} \angle \mathrm{COA} \\ \angle \mathrm{DOE} \mathrm{and} \angle \mathrm{DOB} \end{gathered}$$
$$\begin{gathered} \angle \mathrm{EOD} \mathrm{and} \angle \mathrm{DOA} \\ \angle \mathrm{EOC} \mathrm{and} \angle \mathrm{AOC} \\ \angle \mathrm{AOB} \mathrm{and} \angle \mathrm{BOE} \end{gathered}$$

Question 27:

In Fig., determine the value of x.

Answer 27:

$$\begin{gathered} \angle \mathrm{AOB}+\angle \mathrm{BOC}=180° \left(\mathrm{Linear} \mathrm{pair}\right) \\ \Rightarrow 3x+3x=180° \\ \Rightarrow 6x=180° \\ \Rightarrow x=\frac{180°}{6}=30° \end{gathered}$$

Question 28:

In Fig., AOC is a line, find x.

Answer 28:

$$\begin{gathered} \angle \mathrm{AOB}+\angle \mathrm{BOC}=180° \left(\mathrm{Linear} \mathrm{pair}\right) \\ \Rightarrow 70°+2x=180° \\ \Rightarrow 2x=180°-70°=110° \\ \Rightarrow x=\frac{110°}{2}=55° \end{gathered}$$

Question 29:

In Fig., POS is a line, find x.

Answer 29:

$\angle \mathrm{QOP}+\angle \mathrm{QOR}+\angle \mathrm{ROS}=180°$       (Angles on a straight line)

$$\begin{gathered} \Rightarrow 60°+4x+40°=180° \\ \Rightarrow 100°+4x=180° \\ \Rightarrow 4x=180°-100°=80° \\ \Rightarrow x=\frac{80°}{4}=20° \end{gathered}$$

Question 30:

In Fig., 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 30:

$$\begin{gathered} \angle z=\angle x=45° \left(\mathrm{Vertically} \mathrm{opposite} \mathrm{angles}\right) \\ \mathrm{Now}, \\ \angle x+\angle y=180° \left(\mathrm{Linear} \mathrm{pair}\right) \\ \Rightarrow \angle y=180°-45°=135° \\ \angle u=\angle y=135° \left(\mathrm{Vertically} \mathrm{opposite} \mathrm{angles}\right) \end{gathered}$$

 

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Question 31:

In Fig., 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 31:

$\angle$BOD + $\angle$DOF + $\angle$FOA = 180°        (Linear pair)
∴ $\angle$FOA = $\angle$u = $180°-90°-50°=40°$
$\angle \mathrm{FOA}=\angle x=40°$    (Vertically opposite angles)
$\angle \mathrm{BOD}=\angle z=90°$    (Vertically opposite angles)
$\angle \mathrm{EOC}=\angle y=50°$    (Vertically opposite angles)

Question 32:

In Fig., find the values of x, y and z.

Answer 32:

$$\begin{gathered} \angle y=25° \left(\mathrm{Vertically} \mathrm{opposite} \mathrm{angles}\right) \\ \mathrm{Since} \angle x+\angle y=180° \left(\mathrm{Linear} \mathrm{pair}\right) \\ \therefore \angle x=180°-25°=155° \\ \angle z=\angle x=155° \left(\mathrm{Vertically} \mathrm{opposite} \mathrm{angles}\right) \end{gathered}$$