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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.

$\angle$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.

#### Question 1:

In Fig., line n is a transversal to lines l and m. Identify the following:

(i) Alternate and corresponding angles in Fig. (i).
(ii) Angles alternate to ∠d and ∠g and angles corresponding to angles ∠f and ∠h in Fig. (ii).
(iii) Angle alternate to ∠PQR, angle corresponding to ∠RQF and angle alternate to ∠PQE in Fig. (iii).
(iv) Pairs of interior and exterior angles on the same side of the transversal in Fig. (ii).

(i) Figure (i)
Corresponding angles:
$\angle$EGB and $\angle$GHD
$\angle$HGB and $\angle$FHD
$\angle$EGA and $\angle$GHC
$\angle$AGH and $\angle$CHF
Alternate angles:
$\angle$EGB and $\angle$CHF
$\angle$HGB and $\angle$CHG
$\angle$EGA and $\angle$FHD
$\angle$AGH and $\angle$GHD

(ii) Figure (ii)
Alternate angle to $\angle$d is $\angle$e.
Alternate angle to $\angle$g is $\angle$b.
Also,
Corresponding angle to $\angle$f is $\angle$c.
Corresponding angle to $\angle$h is $\angle$a.

(iii) Figure (iii)
Angle alternate to $\angle$PQR is $\angle$QRA.
Angle corresponding to $\angle$RQF is $\angle$ARB.
Angle alternate to $\angle$POE is $\angle$ARB.

(iv) Figure (ii)
Pair of interior angles are
$\angle$a and $\angle$e
$\angle$d and $\angle$f
Pair of exterior angles are
$\angle$b and $\angle$h
$\angle$c and $\angle$g

#### Question 2:

In Fig., AB and CD are parallel lines intersected by a transversal PQ at L and M respectively. If ∠CMQ = 60°, find all other angles in the figure.

$\angle$ALM = $\angle$CMQ = $60°$        (Corresponding angles)
$\angle$LMD = $\angle$CMQ = $60°$        (Vertically opposite angles)
$\angle$ALM = $\angle$PLB = $60°$          (Vertically opposite angles)
Since
$\angle$CMQ + $\angle$QMD = $180°$     (Linear pair)
QMD = $180°-60°=120°$
$\angle$QMD = $\angle$MLB = $120°$        (Corresponding angles)
$\angle$QMD = $\angle$CML = $120°$        (Vertically opposite angles)
$\angle$MLB = $\angle$ALP = $120°$          (Vertically opposite angles)

#### Question 3:

In Fig., AB and CD are parallel lines intersected by a transversal PQ at L and M respectively. If ∠LMD = 35° find ∠ALM and ∠PLA.

In the given Fig., AB || CD.

#### Question 4:

The line n is transversal to line l and m in Fig. Identify the angle alternate to ∠13, angle corresponding to ∠15, and angle alternate to ∠15.

In this given Fig., line l || m.
Here,
Alternate angle to $\angle$13 is $\angle$7.
Corresponding angle to $\angle$15 is $\angle$7.
Alternate angle to $\angle$15 is $\angle$5.

#### Question 5:

In Fig., line l || m and n is a transversal. If ∠1 = 40°, find all the angles and check that all corresponding angles and alternate angles are equal.

In the given figure, l || m.
Here,

Also,

Thus,

Hence, alternate angles are equal.

#### Question 6:

In Fig., line l || m and a transversal n cuts them at P and Q respectively. If ∠1 = 75°, find all other angles.

In the given figure, l || m, n is a transversal line and ∠1 = 75°.
Thus, we have:

#### Question 7:

In Fig., AB || CD and a transversal PQ cuts them at L and M respectively. If ∠QMD = 100°, find all other angles.

In the given figure, AB || CD, PQ is a transversal line and $\angle$QMD = 100°.
Thus, we have:
$\angle$DMQ + $\angle$QMC = 180°    (Linear pair)
$\therefore \angle \mathrm{QMC}=180°-\angle \mathrm{DMQ}=180°-100°=80°$
Thus,
$\angle$DMQ = $\angle$BLM = 100°         (Corresponding angles)
$\angle$DMQ = $\angle$CML = 100°         (Vertically opposite angles)
$\angle$BLM = $\angle$PLA = 100°           (Vertically opposite angles)
Also,
$\angle$CMQ = $\angle$ALM = 80°         (Corresponding angles)
$\angle$CMQ = $\angle$DML = 80°         (Vertically opposite angles)
$\angle$ALM = $\angle$PLB = 80°           (Vertically opposite angles)

#### Question 8:

In Fig., l || m and p || q. Find the values of x, y, z, t.

In the given figure, l || m and p || q.
Thus, we have:
$\angle z=80°$                (Vertically opposite angles)
$\angle z=\angle t=80°$       (Corresponding angles)
$\angle z=\angle y=80°$       (Corresponding angles)
$\angle x=\angle y=80°$       (Corresponding angles)

#### Question 9:

In Fig., line l || m, ∠1 = 120° and ∠2 = 100°, find out ∠3 and ∠4.

In the given figure, ∠1 = 120° and ∠2 =100°.
Since l || m, so

Also,

We know that the sum of all the angles of triangle is 180°.
$\therefore \angle 6+\angle 3+\angle 4=180°\phantom{\rule{0ex}{0ex}}⇒60°+80°+\angle 4=180°\phantom{\rule{0ex}{0ex}}⇒140°+\angle 4=180°\phantom{\rule{0ex}{0ex}}⇒\angle 4=180°-140°=40°$

#### Question 10:

In Fig., line l || m. Find the values of a, b, c, d. Give reasons.

In the given figure, line l || m.
Thus, we have:

#### Question 11:

In Fig., AB || CD and ∠1 and ∠2 are in the ratio 3 : 2. Determine all angles from 1 to 8.

In the given figure, AB || CD and t is a transversal line.
Now, let:
$\angle 1=3x\phantom{\rule{0ex}{0ex}}\angle 2=2x$
Thus, we have:

Now,

#### Question 12:

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

In the given figure, l || m || n and p is a transversal line.
Thus, we have:

#### Question 13:

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

In the given figure, l || m || n and ∠1 = 60°.
Thus, we have:

#### Question 14:

In Fig., if AB || CD and CD || EF, find ∠ACE.

In the given figure, AB || CD and CD || EF.
Extend line CE to E'.

Thus, we have:

#### Question 15:

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

In the given figure, l || m, n || p and ∠1 = 85°.
Now, let ∠4 be the adjacent angle of ∠2.
Thus, we have:

$\angle 3+\angle 2=180°$       (Sum of interior angles on the same side of the transversal)
$\therefore \angle 2=180°-\angle 3=180°-85°=95°$

#### Question 16:

In Fig., a transversal n cuts two lines l and m. If ∠1 = 70° and ∠7 = 80°, is l || m?

We know that if the alternate exterior angles of two lines are equal, then the lines are parallel.
In the given figure, are alternate exterior angles, but they are not equal.

Therefore, lines l and m are not parallel.

#### Question 17:

In Fig., a transversal n cuts two lines l and m such that ∠2 = 65° and ∠8 = 65°. Are the lines parallel?

$\angle$2 = $\angle$3 = 65°        (Vertically opposite angles)
$\angle$8 = $\angle$6 = 65°         (Vertically opposite angles)
∴ $\angle$3 = $\angle$6
l || m                       (Two lines are parallel if the alternate angles formed with the transversal are equal)

#### Question 18:

In Fig., show that AB || EF.

Extend line CE to E'.

#### Question 19:

In Fig., AB || CD. Find the values of x, y, z.

$\angle x+125°=180°$             (Linear pair)
$\therefore \angle x=180°-125°=55°$

$\angle z=125°$            (Corresponding angles)
$\angle x+\angle z=180°$   (Sum of adjacent interior angles is $180°$)
$\angle x+125°=180°\phantom{\rule{0ex}{0ex}}⇒\angle x=180°-125°=55°$

$\angle x+\angle y=180°$   (Sum of adjacent interior angles is $180°$)
$55°+\angle y=180°\phantom{\rule{0ex}{0ex}}⇒\angle y=180°-55°=125°$

#### Question 20:

In Fig., find out ∠PXR, if PQ || RS.

Draw a line parallel to PQ passing through X.

Here,
(Alternate interior angles)
∵ PQ || RS || XF
∴ $\angle \mathrm{PXR}=\angle \mathrm{PXF}+\angle \mathrm{FXR}=70°+50°=120°$

#### Question 21:

In Fig., we have

(i) ∠MLY = 2 ∠LMQ, find ∠LMQ.
(ii) ∠XLM = (2x − 10)° and ∠LMQ = x + 30°, find x.
(iii) ∠XLM = ∠PML, find ∠ALY
(iv) ∠ALY = (2x − 15)°, and ∠LMQ = (x + 40)°, find x

(i)

(ii)

(iii)

(iv)

#### Question 22:

In Fig., DE || BC. Find the values of x and y.

$\angle$ABC = $\angle$DAB       (Alternate interior angles)

$\angle$ACB = $\angle$EAC       (Alternate interior angles)

#### Question 23:

In Fig., line AC || line DE and ∠ABD = 32°. Find out the angles x and y if ∠E = 122°.

#### Question 24:

In Fig., side BC of ∆ABC has been produced to D and CE || BA. If ∠ABC = 65°, ∠BAC = 55°, find ∠ACE, ∠ECD and ∠ACD.

$\angle$ABC = $\angle$ECD = 55°          (Corresponding angles)
$\angle$BAC = $\angle$ACE = 65°          (Alternate interior angles)
Now, $\angle$ACD = $\angle$ACE + $\angle$ECD
⇒ $\angle$ACD = 55° + 65° = 120°

#### Question 25:

In Fig., line CAAB || line CR and line PR || line BD. Find ∠x, ∠y and ∠z.

Since CA ⊥ AB,
$\therefore \angle x=90°$
We know that the sum of all the angles of triangle is 180°.

$\angle$PBC = $\angle$APQ = $70°$            (Corresponding angles)
Since

#### Question 26:

In Fig., PQ || RS. Find the value of x.

#### Question 27:

In Fig., AB || CD and AE || CF; ∠FCG = 90° and ∠BAC = 120°. Find the values of x, y and z.

$\angle$BAC = $\angle$ACG = 120°          (Alternate interior angle)
∴ $\angle$ACF + $\angle$FCG = 120°
$\angle$ACF = 120° − 90° = 30°

$\angle$DCA + $\angle$ACG = 180°            (Linear pair)
$\angle$x = 180° − 120° = 60°

$\angle$BAC + $\angle$BAE + $\angle$EAC = 360°
$\angle$CAE = 360° − 120° − (60° + 30°) = 150°             ($\angle$BAE =  $\angle$DCF)

#### Question 28:

In Fig., AB || CD and AC || BD. Find the values of x, y, z.

(i) Since AC || BD and CD || AB, ABCD is a parallelogram.
$\angle$CAB + $\angle$ACD = 180°     (Sum of adjacent angles of a parallelogram)
∴ $\angle$ACD = 180° − 65° = 115°
$\angle$CAD = $\angle$CDB = 65°         (Opposite angles of a parallelogram)
$\angle$ACD = $\angle$DBA = 115°       (Opposite angles of a parallelogram)

(ii) Here,
AC || BD and CD || AB
$\angle$DAC = x = 40°            (Alternate interior angle)
$\angle$DAB = y = 35°            (Alternate interior angle)

#### Question 29:

In Fig., state which lines are parallel and why?

Let F be the point of intersection of line CD and the line passing through point E.

Since $\angle$ACD and $\angle$CDE are alternate and equal angles, so
$\angle$ACD = 100° = $\angle$CDE
∴ AC || EF

#### Question 30:

In Fig. 87, the corresponding arms of ∠ABC and ∠DEF are parallel. If ∠ABC = 75°, find ∠DEF.

Construction:
Let G be the point of intersection of lines BC and DE.

∵ AB || DE and BC || EF

$\angle \mathrm{ABC}=\angle \mathrm{DGC}=\angle \mathrm{DEF}=75°$  (Corresponding angles)​

#### Question 1:

Write down each pair of adjacent angles shown in Fig.

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

#### Question 2:

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

In figure (i), the adjacent angles are:

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.

(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

#### Question 4:

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

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°

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

Complementary angles of the following angles are:

#### Question 6:

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

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°

Since

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?

(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.

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.

(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?

Let x and y be two supplementary angles that are equal.
$\angle x=\angle y$
According to the question,
$\angle x+\angle y=180°\phantom{\rule{0ex}{0ex}}⇒\angle x+\angle x=180°\phantom{\rule{0ex}{0ex}}⇒2\angle x=180°\phantom{\rule{0ex}{0ex}}⇒\angle x=\frac{180°}{2}=90°\phantom{\rule{0ex}{0ex}}\therefore \angle x=\angle y=90°$

#### Question 12:

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

Let x be the complement of the given angle $28°$.

So, supplement of the angle = $180°-62°=118°$

#### Question 13:

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

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.

Since OE is the bisector of $\angle$BOD,

#### Question 15:

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

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?

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?

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

#### Question 18:

Can two acute angles form a linear pair?

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.

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

The complement of angle cannot be determined.

#### Question 20:

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

(i)
Since (Linear pair)

(ii)

(iii)

(iv)

(v)
$2x°+x°+2x°+3x°=180°\phantom{\rule{0ex}{0ex}}⇒8x=180\phantom{\rule{0ex}{0ex}}⇒x=\frac{180}{8}=22.5°$

(vi)
$3x°=105°\phantom{\rule{0ex}{0ex}}⇒x=\frac{105}{3}=35°$

#### Question 21:

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

$\angle 1=\angle 3$          (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 $\angle 3=65°$

#### 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?

$\angle$AOC + $\angle$BOC = 180°                   (Linear pair)
$⇒\left(2y+5\right)+3x=180°\phantom{\rule{0ex}{0ex}}⇒3x+2y=175°$
(i) If x = 25°, then
$3×25°+2y=175°\phantom{\rule{0ex}{0ex}}⇒75°+2y=175°\phantom{\rule{0ex}{0ex}}⇒2y=175°-75°=100°\phantom{\rule{0ex}{0ex}}⇒y=\frac{100°}{2}=50°$
(ii) If y = 35°, then
$3x+2×35°=175°\phantom{\rule{0ex}{0ex}}⇒3x+70°=175°\phantom{\rule{0ex}{0ex}}⇒3x=175°-70°=105°\phantom{\rule{0ex}{0ex}}⇒x=\frac{105°}{3}=35°$

#### Question 23:

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

Linear pairs of angles:

#### Question 24:

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

$\angle BOC=x+20°=50°+20°=70°\phantom{\rule{0ex}{0ex}}\angle COD=x=50°\phantom{\rule{0ex}{0ex}}\angle AOD=x+10°=50°+10°=60°\phantom{\rule{0ex}{0ex}}$

#### Question 25:

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

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.?

There are 10 adjacent pairs in the given figure; they are:

#### Question 27:

In Fig., determine the value of x.

#### Question 28:

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

#### Question 29:

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

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

$⇒60°+4x+40°=180°\phantom{\rule{0ex}{0ex}}⇒100°+4x=180°\phantom{\rule{0ex}{0ex}}⇒4x=180°-100°=80°\phantom{\rule{0ex}{0ex}}⇒x=\frac{80°}{4}=20°\phantom{\rule{0ex}{0ex}}$

#### 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.