**Exercise-5.1**

*Q.1. Find the complementary angle of each of the following angles:*

*Ans:*

The sum of values of complementary angles is 90°.

(i) 40°

Complement = 90° – 40° = 50°

(ii) 53°

Complement = 90° – 53° = 37°

(iii) 67°

Complement = 90° – 67° = 23°

*Q2. Find the supplementary angle of each of the following angles:*

** Ans**: The sum of values of supplementary angles is 180°

(1)95°

Supplement = 180° – 95° = 85°

(ii) 83°

Supplement = 180° – 83° = 97°

(iii) 155°

Supplement = 180° – 155° = 25°

*Q3: Find out which of following pairs of angles are supplementary and which are complementary.*

* (i) 60°, 120°*

* (ii) 67°, 23°*

* (iii) 108°, 72°*

* (iv) 150°, 30°*

* (v) 50°, 40°*

* (vi) 72°. 18°*

** Ans:** If the sum of measures of angles is 90°, they are complimentary angles. If the sum of measures of angles is 180°, they are known as supplementary angles.

** (i) 60°, 120°**

Sum of value of the above angles = 60° + 120° =180°

The above angles are supplementary.

** (ii) 67°, 23°**

Sum of value of the above angles = 67° + 23° = 90°

The above angles are complementary.

** (iii) 108°, 72°**

Sum of value of the above angles = 108° + 72° =180°

The above angles are supplementary.

** (iv) 150°, 30°**

Sum of value of the above angles =150° + 30° =180°

The above angles are supplementary.

** (v) 50°, 40°**

Sum of value of the above angles = 50° + 40° = 90°

These angles are complementary angles.

** (vi) 72°, 18°**

Sum of value of the above angles = 72° + 18° = 90°

These angles are complementary angles.

*Q4: Determine the angle that is complimentary by itself.*

** Ans:** Let an angle be a.

Let complement of the angle is also a.

If the sum of measures of angle is 90°, it is complimentary angles

a + a = 90°

2a = 90°

a = 90°/2 = 45°

*Q5. Determine the angle that is supplementary by itself.*

** Ans:** Let an angle be a.

Let the supplement of the angle is also a.

If the sum of measures of angle is 180°, it is known as supplementary angle.

a + a = 180°

2a = 180°

a = 90°

*Q6. In the below given figure, \(\angle 1\) and \(\angle 2\) are supplementary angles. If \(\angle 1\) is decreased, what should be the change \(\angle 2\) so that it remains supplementary?*

** Ans:** \(\angle 1\) and \(\angle 2\) are supplementary angles.

If \(\angle 1\) is decreased, then \(\angle 2 \) should also be increased by the same value so that this angle pair remains supplementary.

*Q7: Can 2 angles be supplementary if both the angles are:*

** (i) Obtuse?**

** (ii) Acute? **

** (iii) Right?**

*Ans:*

(i) No. Obtuse angle will always be greater than 90. Even if we add the minimum supplementary, it will be more than 180°. So, two obtuse angles can’t be a supplementary angle pair.

(ii) No. Because acute angle will always be lesser than 90°. Even if we add the maximum acute angle i.e, 89, it cannot add up to 180°. So, two acute angles cannot make a supplementary angle pair.

(iii) Yes, The value of right angle= 90° and sum of two right angles make 180°

So, 2 right angles together can make a supplementary angle pair.

*Q8: Find out whether the complementary of an angle greater than 45° is lesser than 45° or greater than 45° or equal to 45°.*

** Ans**: Let X and Y are two angles which make complementary angle pair and X is greater than 45°.

X + Y = 90°

Y = 90° – X

Therefore, B will be lesser than 45°.

**Q9. In the following figure:**

**(a) Is \(\angle 1\) adjacent to \(\angle 2\)?**

**(b) Is \(\angle AOC \) adjacent to \(\angle AOE \)?**

**(c) Do \(\angle COE \) and \(\angle EOD \) form a linear pair?**

**(d) Are \(\angle BOD \) and \(\angle DOA \) supplementary?**

**(e) Is \(\angle 1\) vertically opposite to \(\angle 1\)?**

**(f) Which angle is vertically opposite to \(\angle 5\)?**

*Ans:*

(a) Yes. Because they have a vertex O as common and also arm OC as common. Also, their non-common arms, OA and OE are on either side of the common arm.

(b) No. They have vertex O as common and also arm OA as common. However, their non-common arms, OC and OE are on the same side of the common arm. So, these are not adjacent to each other.

(c) Yes. Since they have vertex O as common and arm OE as common. Also, their non-common arms. OC and OD. are opposite rays.

(d) Yes. Since \(\angle BOD\) and \(\angle DOA\) have vertex O as common and their non-common arms are opposite to each other.

(e) Yes. Since these angles are formed by the intersection of two straight lines namely, AB and CD.

(f) \(\angle COB\) is the vertically opposite angle of \(\angle 5\) as these are formed due to the intersection of two straight lines, AB and CD.

*Q10. Looking at the given diagram below, Identify the following pairs of angles*

* (i) Linear pairs.*

* (ii) Vertically opposite angles.*

** Ans:** (i) \(\angle 2\) and \(\angle 1\), \(\angle 1\) and \(\angle 5\) as these have a common vertex and also have non- common arms opposite to each other.

(ii) \(\angle 2\) and \(\angle 5\), \(\angle 1\) and \(\angle 1\) + \(\angle 4\) are vertically opposite angles because these are formed due to intersection of two straight lines.

*Q11. In the below figure, Is \(\angle 1\) adjacent to \(\angle ?\) ?*

* *

* *** Ans:** \(\angle 1\) and \(\angle 2\) are not adjacent angles because they don’t have a common vertex.

*Q12. Find the values of angles A, B and C in each of the following:*

Ans: (i) Since \(\angle A\) and \(\angle 55^{\circ}\) are vertically opposite,

\(\angle A\) = 55 \(^{\circ}\)

\(\angle A\) + \(\angle B\) =\(180^{\circ}\)

\(55^{\circ}\) + \(\angle B\) = \( 180^{\circ}\)

\(\angle B\) = \( 180^{\circ}\) – \( 55^{\circ}\) = \(125^{\circ}\)

\(\angle B\) = \(\angle C\) (Vertically Opposite angles)

\(\angle C\) = \(125^{\circ}\)

(ii) \(\angle C\) = \(40^{\circ}\) (Vertically opposite angles)

\(\angle B\) + \(\angle C\) = \(180^{\circ}\)

\(\angle B\) = \( 180^{\circ}\) – \( 40^{\circ}\) = \(140^{\circ}\)

\( 40^{\circ}\) + \(\angle A\) + \( 25^{\circ}\) = \( 180^{\circ}\)

\( 65^{\circ}\) +\(\angle A\) = \( 180^{\circ}\)

\(\angle A\) = \( 180^{\circ}\) – \( 65^{\circ}\) = \( 115^{\circ}\)

*Q13: Complete the following:*

*(a) The sum of measures of two supplementary angles are ________*

*(b) If 2 adjacent angles are supplementary, they form a ________*

*(c) The sum of measures of two complementary angles are ________ *

*(d) 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*

*(e) The vertically opposite angles are always________, if 2 lines intersect at a point. *

*(f) Two angles forming a linear pair are ________*

*Ans:*

(a) 180°

(b) Linear pair

(c) 90°

(d) Obtuse angles

(e) Equal

(f) Supplementary

*Q14: Name the pairs of angles in the adjoining figure given below.*

(ii) Adjacent complementary angles

(i) Obtuse vertically opposite angles

(iv) Unequal supplementary angles

(v) Adjacent angles that do not form a linear pair

(iii) Equal supplementary angles

**Ans: ** (ii) \(\angle TOP\), \(\angle POQ\)

(i) \(\angle POS\), \(\angle QOR\)

(iv) \(\angle TOP\), \(\angle TOR\)

(v) \(\angle POQ\) and \(\angle POT\), \(\angle POT\) and \(\angle TOS\), \(\angle TOS\) and \(\angle ROS\)

(iii) \(\angle TOQ\), \(\angle TOS\)

__Exercise-5.2__

*Q1: Describe the property that is used in each of the following statements:*

*(i)If \(a||b,\; then\: \angle 1=\angle 5.\).*

*(ii)If \(\angle 4=\angle 6,\, then\: a||b.\)*

*(iii)If \(\angle 4+\angle 5+180^{\circ},\, then\: a||b.\)*

**Ans:**

(i)Given, \(a||b\), then \(\angle 1=\angle 5\) [Corresponding angles]

If two parallel lines are cut by a transversal, each pair of corresponding angles are equal in measure.

(ii) Given, \(\angle 4=\angle 6\), then \(a||b\) [Alternate angles]

When a transversal cuts two lines such that pairs of alternate interior angles are equal, the lines have to be parallel.

(iii)Given, \(\angle 4+\angle 5=180^{\circ}\), then \(a||b\) [Co-interior angles]

When a transversal cuts two lines, such that pairs of interior angles on the same side of transversal are supplementary, the lines have to be parallel.

*Q2: 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.*

*(iv)The vertically opposite angles.*

**Ans:**

(i)The pairs of corresponding angles:

\(\angle 1,\angle 5;\angle 2,\angle 6;\angle 4,\angle 8 and \angle 3,\angle 7\)(ii)The pairs of alternate interior angles are:

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

\(\angle 3,\angle 8 \: and\: \angle 2,\angle 5\)(iv)The vertically opposite angles are:

\(\angle 1,\angle 3 ;\angle 2,\angle 4;\angle 6,\angle 8 \: and\: \angle 5,\angle 7\)

*Q3: In the adjoining figure, \(p||q\). Find the unknown angles.*

**Ans:** Given, \(p||q\) and cut by a transversal line.

\(\ because 125^{\circ}+e=180^{\circ}\) [Linear Pair]

\(\ therefore 1e=180^{\circ}-125^{\circ}=55^{\circ}\) ……(1)

Now \(e=f=55^{\circ}\) [Vertically opposite angles]

Also \(a=f=55^{\circ}\) [Alternate interior angles]

\(a+b=180^{\circ}\) [Linear pair]

\(\Rightarrow 55^{\circ}+b=180^{\circ}\) [From equation (1)]

\(\Rightarrow b=180^{\circ}-55^{\circ}=125^{\circ}\)Now \(a=c=55^{\circ}\) and \(b=d=125^{\circ}\) [Vertically opposite angles]

Thus, \(a=55^{\circ},b=125^{\circ},c=55^{\circ},d=125^{\circ},e=55^{\circ} \: and \: f=55^{\circ}\)

*Q4: Find the values of x in each of the following figures if \(l||m\)*

**Ans:**

(i)Given, \(l||m\) and t is transversal line.

\(\ therefore\) Interior vertically opposite angle between lines l and t=\(110^{\circ}\)

\(\ therefore 110^{\circ}+x=180^{\circ}\) [Supplementary angles]

\(\Rightarrow x=180^{\circ}-110^{\circ}=70^{\circ}\)(ii)Given, \(l||m\) and t is transversal line.

\(x+2x=180^{\circ}\\ \\ \Rightarrow 3x=180^{\circ} [Interior \;opposite\; angles]\\ \\ \Rightarrow x=\frac{180^{\circ}}{3}=60^{\circ}\)(iii)Given, \(l||m\) and \(a||b\).

\(x=100^{\circ}\) [Corresponding angles]

*Q5: In the given figure, the arms of two angles are parallel. If \(\bigtriangleup ABC=70^{\circ}\), then find:*

*(i) \(\angle DGC\)*

*(II) \(\angle DEF\)*

**Ans:**

(i) From the figure \(AB\parallel DE\) and BC is a transversal line and \(\angle ABC=70^{\circ}\)

\(\ because \angle ABC=\angle DGC\) (corresponding angles)

\(∴ \angle DGC=70^{\circ}\) ………(i)

(ii) From the figure \(BC\parallel DE\) and DE is a transversal line and \(\angle DGC=70^{\circ}\)

\(\ because \angle DGC=\angle DEF\) (corresponding angles)

\(∴ \angle DGC=70^{\circ}\) ………(i)

*Q6: In the given figure below, decide whether \(l\) is parallel to \(m\).*

**Ans:**

(i) \(126^{\circ}+44^{\circ}=170^{\circ}\)

Here \(l\parallel m\) this condition holds false, because the sum of interior opposite angles are not equal to \(180^{\circ}\).

(ii) \(75^{\circ}+75^{\circ}=150^{\circ}\)

Here \(l\parallel m\) this condition holds false, because the sum of interior opposite angles are not equal to \(180^{\circ}\).

(iii) \(57^{\circ}+123^{\circ}=180^{\circ}\)

Here \(l\parallel m\) this condition holds true, because the sum of interior opposite angles are equal to \(180^{\circ}\).

(iv) \(98^{\circ}+72^{\circ}=170^{\circ}\)

Here \(l\parallel m\) this condition holds false, because the sum of interior opposite angles are not equal to \(180^{\circ}\).