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LINES AND ANGLES
Exercise 5.1
Q.1. Find the complement of each of the following angles :
(i)
(ii)
(iii)
Ans. We know that sum of the measures of two complementary
angles is 90°.
(i)
Let other angle = x
As per condition, x + 20° = 90°
⇒
x = 90° – 20°
or
x = 70°
Hence, complement angle of 20° is 70°
(ii)
Let other angle = x
As per condition, x + 63° = 90°
⇒
x = 90° – 63°
or
x = 27°
Hence, complement angle of 63° is 27°
(iii)
Let other angle = x
As per condition, x + 57° = 90°
⇒
x = 90° – 57°
or
x = 33°
Hence, complement angle of 57° is 33°
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Q.2. Find the supplement of each of the following angles :
(i)
(ii)
(iii)
Ans. We know that sum of the measures of two supplementary
angles is 180°.
(i)
Let other angle = x
As per condition, x + 105° = 180°
⇒
x = 180° – 105°
or
x = 75°
Hence, supplement angle of 105° is 75°
(ii)
Let other angle = x
As per condition, x + 87° = 180°
⇒
x = 180° – 87°
or
x = 93°
Hence, supplement angle of 87° is 93°.
(iii)
Let other angle = x
As per condition, x + 154 ° = 180°
⇒
x = 180° – 154° = 26°
Hence, supplement angle of 154° is 26°
Q.3. Identify which of the following pairs of angles are
complementary and which are supplementary.
(i) 65°, 115°
(ii) 63°, 27°
(iii) 112°, 68°
(iv) 130°, 50°
(v) 45°, 45°
(vi) 80°, 10°
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Ans. We know that when the sum of two angles is 90°, the angles
are known as complementary angles.
When the sum of two angles is 180°, the angles are known
as supplementary angles.
(i) 65°, 115°
⇒
(given)
Sum = 65° + 115° = 180°
Hence, given angles are supplementary angles.
(ii)
63°, 27°
⇒
(given)
Sum = 63° + 27° = 90°
Hence, given angles are complementary angles.
(iii) 112°, 68°
⇒
(given)
Sum = 112° + 68° = 180°
Hence, given angles are supplementary angles.
(iv) 130°, 50°
⇒
(given)
Sum = 130° + 50° = 180°
Hence, given angles are supplementary angles.
(v)
45°, 45°
⇒
(given)
Sum = 45° + 45° = 90°
Hence, given angles are complementary angles.
(vi)
80°, 10°
⇒
(given)
Sum = 80° + 10° = 90°
Hence, given angles are complementary angles.
Q.4. Find the angle which is equal to its complement.
Ans.
Let the angle be x.
So, complement of x = 90° – x
As per question, x = 90° – x
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⇒
x + x = 90°
or
2x = 90°
or
x = 45°
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Hence, the angle which is equal to its complement is 45°.
Q.5. Find the angle which is equal to its supplement.
Ans.
Let the angle be x.
So, supplement of x = 180° – x
As per condition, x = 180° – x
∴
x + x = 180°
or
2x = 180°
∴
x = 90°
∴ The angle which is equal to its supplement is 90°.
Q.6. In the given figure, ∠1, and ∠2 are supplementary
angles. If ∠1 is decreased, what changes
should take place in ∠2 so that both the
angles still remain supplementary?
Ans. ∠1 + ∠2 = 180° (Supplementary angles)
If ∠1 is decreased, then ∠2 must be increased
to maintain the sum of angles equal to 180°.
Q.7. Can two angles be supplementary if both of them are :
(i) acute
(ii) obtuse
(iii) right?
Ans. We know that the sum of the measures of two
supplementary angles is 180°.
(i) Measure of an acute angle < 90°
⇒
Sum of the measures of two acute angles is less
than 180°.
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Hence, two acute angles cannot be the supplement
of each other.
(ii) Measure of an obtuse angle > 90°
⇒
Sum of the measures of two obtuse angles is
greater than 180°.
Hence, two obtuse angles cannot be the
supplement of each other.
(iii) Measure of a right angle = 90°
⇒
Sum of the measures of two right angles = 180°.
Hence, two right angles can be the supplement of
each other.
Q.8. An angle is greater than 45°. Is its complementary angle
greater than 45° or equal to 45° or less than 45°?
Ans. Two angles are said to be complementary if their sum
is 90°. If one angle is greater than 45°, then its complement
will be obviously less than 45°.
Q.9. In the adjoining figure :
(i) Is ∠1 adjacent to ∠2?
(ii) Is ∠AOC adjacent to ∠AOE?
(iii) Do ∠COE and ∠EOD form a
linear pair?
(iv) Are ∠BOD and ∠DOA supplementary?
(v) Is ∠1 vertically opposite to ∠4?
(vi) What is the vertically opposite angle of ∠5?
∠AOC = ∠1
Ans. (i)
∠EOC = ∠2
⇒
OC = OC [Common arm and O is
common vertex.]
Hence, ∠1 and ∠2 are adjacent angles.
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(ii) ∠AOC and ∠AOE
∠AOE = ∠AOC + ∠EOC
∠AOE is sum of ∠AOC and ∠EOC.
Hence, ∠AOC and ∠AOE are not adjacent angles.
(iii) In ∠COE and ∠EOD
OE is common arm and O is common vertex. The
non-common arms OC and OD form opposite rays.
Hence, ∠COE and ∠EOD form a linear pair.
(iv) ∠BOD and ∠DOA form linear pair.
So, ∠BOD + ∠DOA = 180°
Hence, we can say that ∠BOD and ∠DOA are
supplementary.
(v) Yes, ∠1 and ∠4 have only common vertex O but no
common arms.
Hence, they are vertically
opposite
angles formed by
HJJG
HJJ
G
intersecting lines AB and CD
(vi) ∠5 = ∠AOD
Vertically opposite angle of ∠AOD = ∠BOC
Q.10. Indicate which pairs of angles are :
(i) Vertically opposite angles.
(ii) Linear pairs.
Ans. (i) Vertically opposite angles are :
∠1 and ∠4 and ∠5 and (∠2 + ∠3)
(ii) ∠4 + ∠5 = 180° and ∠1 + ∠5 = 180°
Hence, linear pairs are (∠4, ∠5) and (∠1, ∠5)
Q.11. In the following figure, is ∠1 adjacent to ∠2? Give
reasons.
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Ans. ∠1 and ∠2 are not adjacent angles because they have no
common vertex.
Q.12. Find the values of the angles x, y and z in each of the
following :
∠x = 55° (Vertically opposite angles)
Ans. (i)
∠x + ∠y = 180° (Linear pair)
So, 55° + ∠y = 180°
∴
∠y = 180° – 55°
⇒
∠y = 125°
Since ∠y = ∠ z (Vertically opposite angles)
So, ∠z = 125°
Hence,
x = 55°, y = 125° and z = 125°
(ii) ∠40° + ∠x + ∠25° = 180°
⇒ ∠x + 65° = 180°
⇒
∠x = 180° – 65° = 115°
∠z = 40° (Vertically opposite angles)
⇒
∠y + ∠z = 180° (Linear pair)
So, ∠y + 40° = 180°
⇒
∠y = 180° – 40° = 140°
Hence, x = 115°, y = 140° and z = 40°
Q.13. Fill in the blanks :
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(i) If two angles are complementary, then the sum of
their measures is..............
(ii) If two angles are supplementary, then the sum of
their measures is................
(iii) Two angles forming a linear pair are...............
(iv) If two adjacent angles are supplementary, they form
a ……………..
(v) If two lines intersect at a point, then the vertically
opposite angles are always...............
(vi) If two lines intersect at a point, and if one pair of
vertically opposite angles are acute angles, then the
other pair of vertically opposite angles are..............
90°
(ii) 180°
(iii) supplementary
(iv) linear pair
(v) equal
(vi) obtuse angles
Ans. (i)
Q.14. In the adjoining figure, name the following pairs of
angles.
(i) Obtuse vertically opposite angles
(ii) Adjacent complementary angles
(iii) Equal supplementary angles
(iv) Unequal supplementary angles
(v) Adjacent angles that do not form a linear pair.
Ans. (i)
∠BOC and ∠AOD
(ii)
∠AOB and ∠AOE
(iii)
∠BOE and ∠EOD
(iv) (∠EOA and ∠EOC) and (∠AOB and ∠BOC)
(v) (∠COD and ∠DOE), (∠DOE and ∠EOA), and
(∠BOA and ∠AOE)
Exercise 5.2
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Q.1. State the property that is used in each of the following
statements?
(i) If a || b, then ∠1 = ∠5.
(ii) If ∠4 = ∠6, then a || b.
(iii) If ∠4 + ∠5 = 180°, then a || b.
Ans.(i)
Since, a || b any pair of corresponding
angles formed when a transversal
intersects a and b are equal.
So, ∠1 = ∠5
(ii) When a transversal cuts two lines, such that pairs of
alternate interior angles are equal, the lines are parallel.
(iii) When a transversal cuts two lines, such that pairs of
interior angles on the same side of the transversal are
supplementary, the lines are parallel.
Q.2. In the adjoining figure, identify
(i) the pairs of corresponding angles.
(ii) the pairs of alternate interior angles.
(iii) the pairs of interior angles on the
same side of the transversal.
(iv) the vertically opposite angles.
Ans. (i) ∠1 and ∠5, ∠4 and ∠8, ∠3 and ∠7, ∠2 and ∠6.
(ii) ∠3 and ∠5, ∠2 and ∠8
(iii) ∠2 and ∠5, ∠3 and ∠8
(iv) ∠1 and ∠3, ∠4 and ∠2,
∠8 and ∠6, ∠5 and ∠7.
Q.3. In the adjoining figure, p || q. Find the
unknown angles.
Ans. (i) ∠f + ∠125° = 180°
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(Linear pair)
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∴
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∠f = 180° – 125° = 55°
∠e = ∠f = 55° (Vertically opposite angles)
∠d = 125° (Corresponding angles)
∠d = ∠b = 125° (Vertically opposite angles)
∠e = ∠a = 55° (Corresponding angles)
∠a = ∠c = 55° (Vertically opposite angles)
Hence, a = 55°, b = 125°, c = 55°, d = 125°, e = 55°
and f = 55°.
Q.4. Find the value of x in each of the following figures, if
l || m.
Ans. (i) In the figure,
∠1 + 110° = 180° [Linear pair]
∴
∠1 = 180° – 110° = 70°
and x = ∠1 [Corresponding angles]
= 70°
(ii) Since, l || m
x = 100° (Corresponding angles)
Q.5. In the given figure, the arms of two angles are parallel.
If ∠ABC = 70°, then find
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(i) ∠DGC
(ii) ∠DEF
Ans. Since AB || DE
∠ABC = ∠DGC (Corresponding angles)
So, ∠DGC = 70°
Since BC || EF,
∠DGC = ∠DEF = 70° (Corresponding angles)
Hence, (i)∠DGC = 70°
(ii) ∠DEF = 70°
Q.6. In the given figures below, decide whether l is parallel
to m.
Ans. (i) ∠126° + ∠44° = 170° ≠ 180°
The interior angles on the same side of the transversal are
not supplementary,
Hence l is not parallel to m.
(ii) l is not parallel to m because alternate angles are not
equal.
(iii)
∠57° + x = 180° (Linear pair)
∴
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x = 180° – 57° = 123°
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Now
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x = 123° = (Alternate angles)
Hence, l || m because alternate angles are equal.
(iv) x + 72° = 180° (Linear pair)
⇒ x = 180° – 72° = 108°
But, x and 98° are alternate angles.
Hence, l is not parallel to m because
alternate angles are not equal.
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