(i) Complement of 20° = 90° – 20° = 70°

(ii) Complement of 63° = 90° – 63° = 27°

(iii) Complement of 57° = 90° – 57° = 33°

(i) Supplement of 105° = 180° – 105° = 75°

(ii) Supplement of 87° = 180° – 87° = 93°

(iii) Supplement of 154° = 180° – 154° = 26°

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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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(i) 65° (+) 115° = 180°

They are supplementary angles.

(ii) 63° (+) 27° = 90°

They are complementary angles.

(iii) 112° (+) 68° = 180°

They are supplementary angles.

(iv) 130° (+) 50° = 180°

They are supplementary angles.

(v) 45° (+) 45° = 90°

They are complementary angles.

(vi) 80° (+) 10° = 90°

They are complementary angles.

Let the required angle be x°.

its complement = (90 – x)°

According to question,

x= 90 – x
Or, x + x = 90
Or, 2x = 90
Or, x = 90/2 = 45°
Thus the required angles are 45°.

Let the required angle be x°.

So, it supplement = (180 – x)°

Now, x = 180 – x

Or, x + x = 180

Or, 2x = 180°

Or x=180°/2=90°

Thus, the required angle is 90°.

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Q.6 In the given figure, \(\angle \) 1 and \(\angle \)2 are supplementary angles.
If \(\angle \) 1 is decreased, what changes should take place in \(\angle \) 2 so that both the angles still remain
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\(\angle \) 1 + \(\angle \)2 = 180° (given)

If \(\angle \) 1 is decreased by some degrees, then angle 2 will also be increased by the same degree so that the two angles still remain supplementary.

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Q.7 Can two angles be supplementary if both of them are:
(i) acute?
(ii) obtuse?
(iii) right?
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(i) No, If two angles are acute, means less than 90°, the two angles cannot be supplementary. Because, their sum will be always less than 180° .

(ii) No. If two angles are obtuse, means more than 90°, the two angles cannot be supplementary. Because, their sum will be always more than 180°.

(iii) Yes. If two angles are right, means both measures 90°, then two angles can form a supplementary pair.

90° + 90° = 180°

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Q.8 An angle is greater than 45°. Is its complementary angle greater than 45° or equal to 45° or less than 45 °?
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Given angle is greater than 45°

Let the given angle be x°.

So, x > 45

Complement of x° = 90° – x° < 45° [ because x > 45°]

Thus the required angle is less than 45°.

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Q.9 In the following figure:
(i) Is \(\angle \)1 adjacent to angle 2?
(ii) Is \(\angle \)AOC adjacent to angle AOE?
(iii) Do \(\angle \) COE and \(\angle \) EOD form a linear pair?
(iv) Are \(\angle \) BOD and \(\angle \) DOA supplementary?
(v) Is \(\angle \) 1 vertically opposite angle to angle 4?
(vi) What is the vertically opposite angle of \(\angle \) 5?
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(i) Yes, \(\angle \) 1 and \(\angle \) 2 are adjacent angles.

because it's one arm (OC) is common

(ii) No, \(\angle \) AOC is not adjacent to \(\angle \) AOE.

[ because OC and OE do not lie on either side of common arm OA] .

(iii) Yes, \(\angle \) COE and \(\angle \) EOD form a linear pair of angles.

(iv) Yes, \(\angle \) BOD and \(\angle \) DOA are supplementary.

[Because \(\angle \) BOD + because \(\angle \)DOA = 180°]

(v) Yes, \(\angle \) 1 is vertically opposite to \(\angle \) 4.

(vi) Vertically opposite angle of \(\angle \) 5 is \(\angle \) 2 + \(\angle \) 3 i.e. angle BOC.

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Q.10 Indicate which pairs of angles are:
(i) Vertically opposite angles
(ii) Linear pairs
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(i) Vertically opposite angles are angle1 and angle 4,

\(\angle \) 5 and (\(\angle \) 2 + \(\angle \) 3)

(ii) Linear pairs are

\(\angle \) 1 and \(\angle \) 5, \(\angle \) 5 and \(\angle \) 4

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Q.11 In the following figure, is \(\angle \)1 adjacent to \(\angle \)2? Give reasons.
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No, \(\angle \) 1 and \(\angle \) 2 are not adjacent

Reasons:

(i) They have no common vertex.

(ii) \(\angle \)1 + \(\angle \)2 is not equals180°

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Q.12 Find the values of the angles x, y and z in each of the following:
(i)
(ii)
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From Fig. 1. we have

\(\angle \)x = 55° (Vertically opposite angles)

\(\angle \) x + \(\angle \) y = 180° (Adjacent angles)

55° + \(\angle \) y = 180° (Linear pair angles)

So, \(\angle \) y = 180° – 55° = 125°

\(\angle \) y = \(\angle \) z (Vertically opposite angles)

\(\angle \) z = 125 °

Hence, \(\angle \) x = 55°, \(\angle \) y = 125° and \(\angle \) z = 125°

From Fig. ii . we have

25° + x + 40° = 180° (Sum of adjacent angles on straight line)

65° + x = 180°

So, x = 180° – 65° = 115°

40° + y = 180° (Linear pairs)

So, y = 180° – 40° = 140°

y + z = 180° (Linear pairs)

140° + z = 180°

So, z = 180° – 140° = 40°

Hence, x – 115°, y = 140° and z – 40°

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Q.13 Fill in the blanks:
(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 ______ .
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(i) If two angles are complementary, then the sum of their measures is _90°_ .

(ii) If two angles are supplementary, then the sum of their measures is _180°_ .

(iii) Two angles forming a linear pair are _suplimentry_ .

(iv) If two adjacent angles are supplementary, they form a _linear pair_.

(v) If two lines intersect at a point, then the vertically opposite angles are always _equal_ .

(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 _obtuse angle_ .

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Q.14 In the given 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 but do not form a linear pair.
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(i) \(\angle \)BOC and \(\angle \)AOD are obtuse vertically opposite angles.

(ii) \(\angle \)AOB and \(\angle \)AOE are adjacent complementary angles.

(iii) \(\angle \)EOB and \(\angle \)EOD are equal supplementary angles.

(iv) \(\angle \)EOA and \(\angle \)EOC are unequal supplementary angles.

(v) \(\angle \)AOB and \(\angle \)AOE, \(\angle \)AOE and \(\angle \)EOD, \(\angle \)EOD and \(\angle \)COD are adjacent angles but do not form a linear pair.

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Q.1 State the property that is used in each of the following statements?
(i) If a \(\parallel \) b, then \(\angle \)1 = \(\angle \)5
(ii) If \(\angle \)4 = \(\angle \)6, then a \(\parallel \) b
(iii) If \(\angle \)4 + \(\angle \)5 = 180°, then a \(\parallel \) b
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(i) Given a \(\parallel \)

So, \(\angle \)1 = \(\angle \)5 (Pair of corresponding angles)

(ii) Given: \(\angle \)4 = \(\angle \)6

So, a \(\parallel \) b [If pair of alternate angles are equal, then the lines are parallel]

(iii) Given: \(\angle \)4 + \(\angle \)5 = 180°

So, a \(\parallel \) b [If sum of interior angles is 180°, then the lines are parallel]

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Q.2 In the given 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.
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(i) The pair of corresponding angles are \(\angle \)1 and \(\angle \)5, \(\angle \)2 and \(\angle \)6, \(\angle \)4 and \(\angle \)8, \(\angle \)3 and \(\angle \)7.

(ii) The pairs of alternate interior angles are \(\angle \)2 and \(\angle \)8, \(\angle \)3 and \(\angle \)5.

(iii) The pairs of interior angles on the same side of the transversal are \(\angle \)2 and \(\angle \)5, \(\angle \)3 and \(\angle \)8.

\(\angle \)e + 125° = 180° (Linear pair)

\(\therefore \) \(\angle \)e = 180° – 125° = 55°

\(\angle \)e = \(\angle \)f (Vertically opposite angles)

\(\therefore \) \(\angle \)f= 55°

\(\angle \)a = \(\angle \)f= 55° (Alternate interior angles)

\(\angle \)c = \(\angle \)a = 55° (Vertically opposite angles)

\(\angle \)d = 125° (Corresponding angles)

\(\angle \)b = \(\angle \)d = 125° (Vertically opposite angles)

Thus, \(\angle \)a = 55°, \(\angle \)b = 125°, \(\angle \)c = 55°, \(\angle \)d = 125°, \(\angle \)e = 55°, \(\angle \)f= 55°.

(i) Let the angle opposite to 110° be y.

\(\therefore \) y = 110° (Vertically opposite angles)

\(\angle \)x + \(\angle \)y = 180° (Sum of interior angle on the same side of transversal)

\(\angle \)x + 110° = 180° .

\(\therefore \) \(\angle \)x = 180° – 110° = 70°

Thus x= 70°

(ii) \(\angle \)x = 110° (Pair of corresponding angles)

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Q.5 the given figure, the arms of two angles are parallel. If \(\angle \)ABC = 70°, then find
(i) \(\angle \)DGC
(ii) \(\angle \)DEF
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Given

AB \(\parallel \) DE

BC \(\parallel \) EF

\(\angle \)ABC = 70°

\(\angle \)DGC = \(\angle \)ABC

(i) \(\angle \)DGC = 70° (Pair of corresponding angles)

\(\angle \)DEF = \(\angle \)DGC

(ii) \(\angle \)DEF = 70° (Pair of corresponding angles)

(i) Sum of interior angles on the same side of transversal
= 126° + 44° = 170° \(\neq \)180°

\(\therefore \) l is not parallel to m.

(ii) Let angle opposite to 75° be x.

x = 75° [Vertically opposite angles]

\(\therefore \) Sum of interior angles on the same side of transversal

= x + 75° = 75° + 75°

= 150° \(\neq \) 180°

\(\therefore \) l is not parallel to m.

(iii) Let the angle opposite to 57° be y.

\(\therefore \) \(\angle \)y = 57° (Vertically opposite angles)

\(\therefore \) Sum of interior angles on the same side of transversal

= 57° + 123° = 180°

\(\therefore \) l is parallel to m.

(iv) Let angle opposite to 72° be z.

\(\therefore \)z = 70° (Vertically opposite angle)

[Sum of interior angles on the same side of transversal]

= z + 98° = 72° + 98°= 170° \(\neq \) 180°

\(\therefore \) l is not parallel to m.