# Chapter – 5: Lines and Angles

# EXERCISE 5.1

**1. Find the complement of each of the following angles:**

**(i)**

Two angles are said to be complementary if the sum of their measures is 90^{o}.

The given angle is 20^{o}

Let the measure of its complement be x^{o}.

Then,

= x + 20^{o} = 90^{o}

= x = 90^{o} – 20^{o}

= x = 70^{o}

Hence, the complement of the given angle measures 70^{o}.

**(ii)**

**Solution: –**

Two angles are said to be complementary if the sum of their measures is 90^{o}.

The given angle is 63^{o}

Let the measure of its complement be x^{o}.

Then,

= x + 63^{o} = 90^{o}

= x = 90^{o} – 63^{o}

= x = 27^{o}

Hence, the complement of the given angle measures 27^{o}.

**(iii)**

**Solution: –**

Two angles are said to be complementary if the sum of their measures is 90^{o}.

The given angle is 57^{o}

Let the measure of its complement be x^{o}.

Then,

= x + 57^{o} = 90^{o}

= x = 90^{o} – 57^{o}

= x = 33^{o}

Hence, the complement of the given angle measures 33^{o}.

**2. Find the supplement of each of the following angles:**

**(i)**

**Solution: –**

Two angles are said to be supplementary if the sum of their measures is 180^{o}.

The given angle is 105^{o}

Let the measure of its supplement be x^{o}.

Then,

= x + 105^{o} = 180^{o}

= x = 180^{o} – 105^{o}

= x = 75^{o}

Hence, the supplement of the given angle measures 75^{o}.

**(ii)**

**Solution: –**

Two angles are said to be supplementary if the sum of their measures is 180^{o}.

The given angle is 87^{o}

Let the measure of its supplement be x^{o}.

Then,

= x + 87^{o} = 180^{o}

= x = 180^{o} – 87^{o}

= x = 93^{o}

Hence, the supplement of the given angle measures 93^{o}.

**(iii)**

**Solution: –**

Two angles are said to be supplementary if the sum of their measures is 180^{o}.

The given angle is 154^{o}

Let the measure of its supplement be x^{o}.

Then,

= x + 154^{o} = 180^{o}

= x = 180^{o} – 154^{o}

= x = 26^{o}

Hence, the supplement of the given angle measures 93^{o}.

**3. Identify which of the following pairs of angles are complementary and which are supplementary.**

**(i) 65**^{o}**, 115**^{o}

**Solution: –**

We have to find the sum of given angles to identify whether the angles are complementary or supplementary.

Then,

= 65^{o} + 115^{o}

= 180^{o}

If the sum of two angle measures is 180^{o}, then the two angles are said to be supplementary.

∴These angles are supplementary angles.

**(ii) 63**^{o}**, 27**^{o}

**Solution: –**

We have to find the sum of given angles to identify whether the angles are complementary or supplementary.

Then,

= 63^{o} + 27^{o}

= 90^{o}

If the sum of two angle measures is 90^{o}, then the two angles are said to be complementary.

∴These angles are complementary angles.

**(iii) 112**^{o}**, 68**^{o}

**Solution: –**

We have to find the sum of given angles to identify whether the angles are complementary or supplementary.

Then,

= 112^{o} + 68^{o}

= 180^{o}

If the sum of two angle measures is 180^{o}, then the two angles are said to be supplementary.

∴These angles are supplementary angles.

**(iv) 130**^{o}**, 50**^{o}

**Solution: –**

Then,

= 130^{o} + 50^{o}

= 180^{o}

If the sum of two angle measures is 180^{o}, then the two angles are said to be supplementary.

∴These angles are supplementary angles.

**(v) 45**^{o}**, 45**^{o}

**Solution: –**

Then,

= 45^{o} + 45^{o}

= 90^{o}

If the sum of two angle measures is 90^{o}, then the two angles are said to be complementary.

∴These angles are complementary angles.

**(vi) 80**^{o}**, 10**^{o}

**Solution: –**

Then,

= 80^{o} + 10^{o}

= 90^{o}

If the sum of two angle measures is 90^{o}, then the two angles are said to be complementary.

∴These angles are complementary angles.

**4. Find the angles which is equal to its complement.**

**Solution: –**

Let the measure of the required angle be x^{o}.

We know that, sum of measures of complementary angle pair is 90^{o}.

Then,

= x + x = 90^{o}

= 2x = 90^{o}

= x = 90/2

= x = 45^{o}

Hence, the required angle measures is 45^{o}.

**5. Find the angles which is equal to its supplement.**

**Solution: –**

Let the measure of the required angle be x^{o}.

We know that, sum of measures of supplementary angle pair is 180^{o}.

Then,

= x + x = 180^{o}

= 2x = 180^{o}

= x = 180/2

= x = 90^{o}

Hence, the required angle measures is 90^{o}.

**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 angles still remain supplementary.**

**Solution: –**

From the question, it is given that,

∠1 and ∠2 are supplementary angles.

If ∠1 is decreased, then ∠2 must be increased by the same value. Hence, this angle pair remains supplementary.

**7. Can two angles be supplementary if both of them are:**

**(i). Acute?**

**Solution: –**

No. If two angles are acute, means less than 90^{o}, the two angles cannot be supplementary. Because, their sum will be always less than 90^{o}.

**(ii). Obtuse?**

**Solution: –**

No. If two angles are obtuse, means more than 90^{o}, the two angles cannot be supplementary. Because, their sum will be always more than 180^{o}.

**(iii). Right?**

**Solution: –**

Yes. If two angles are right, means both measures 90^{o}, then two angles can form a supplementary pair.

∴90^{o }+ 90^{o} = 180

**8. An angle is greater than 45**^{o}**. Is its complementary angle greater than 45**^{o}** or equal to 45**^{o}** or less than 45**^{o}**?**

**Solution: –**

Let us assume the complementary angles be p and q,

We know that, sum of measures of complementary angle pair is 90^{o}.

Then,

= p + q = 90^{o}

It is given in the question that p > 45^{o}

Adding q on both the sides,

= p + q > 45^{o }+ q

= 90^{o} > 45^{o }+ q

= 90^{o} – 45^{o} > q

= q < 45^{o}

Hence, its complementary angle is less than 45^{o}.

**9. In the adjoining figure:**

**(i) Is ****∠****1 adjacent to ****∠****2?**

**Solution: –**

By observing the figure we came to conclude that,

Yes, as ∠1 and ∠2 having a common vertex i.e. O and a common arm OC.

Their non-common arms OA and OE are on both the side of common arm.

**(ii) Is ****∠****AOC adjacent to ****∠****AOE?**

**Solution: –**

By observing the figure, we came to conclude that,

No, since they are having a common vertex O and common arm OA.

But, they have no non-common arms on both the side of the common arm.

**(iii) Do ****∠****COE and ****∠****EOD form a linear pair?**

**Solution: –**

By observing the figure, we came to conclude that,

Yes, as ∠COE and ∠EOD having a common vertex i.e. O and a common arm OE.

Their non-common arms OC and OD are on both the side of common arm.

**(iv) Are ****∠****BOD and ****∠****DOA supplementary?**

**Solution: –**

By observing the figure, we came to conclude that,

Yes, as ∠BOD and ∠DOA having a common vertex i.e. O and a common arm OE.

Their non-common arms OA and OB are opposite to each other.

**(v) Is ****∠****1 vertically opposite to ****∠****4?**

**Solution: –**

Yes, ∠1 and ∠2 are formed by the intersection of two straight lines AB and CD.

**(vi) What is the vertically opposite angle of ****∠****5?**

**Solution: –**

∠COB is the vertically opposite angle of ∠5. Because these two angles are formed by the intersection of two straight lines AB and CD.

**10. Indicate which pairs of angles are:**

**(i) Vertically opposite angles.**

**Solution: –**

By observing the figure we can say that,

∠1 and ∠4, ∠5 and ∠2 + ∠3 are vertically opposite angles. Because these two angles are formed by the intersection of two straight lines.

**(ii) Linear pairs.**

**Solution: –**

By observing the figure we can say that,

∠1 and ∠5, ∠5 and ∠4 as these are having a common vertex and also having non common arms opposite to each other.

**11. In the following figure, is ∠1 adjacent to ∠2? Give reasons.**

**Solution: –**

∠1 and ∠2 are not adjacent angles. Because, they are not lie on the same vertex.

**12. Find the values of the angles x, y, and z in each of the following:**

**(i)**

**Solution: –**

∠x = 55^{o}, because vertically opposite angles.

∠x + ∠y = 180^{o} … [∵ linear pair]

= 55^{o} + ∠y = 180^{o}

= ∠y = 180^{o} – 55^{o}

= ∠y = 125^{o}

Then, ∠y = ∠z … [∵ vertically opposite angles]

∴ ∠z = 125^{o}

**(ii)**

**Solution: –**

∠z = 40^{o}, because vertically opposite angles.

∠y + ∠z = 180^{o} … [∵ linear pair]

= ∠y + 40^{o} = 180^{o}

= ∠y = 180^{o} – 40^{o}

= ∠y = 140^{o}

Then, 40 + ∠x + 25 = 180^{o} … [∵angles on straight line]

65 + ∠x = 180^{o}

∠x = 180^{o} – 65

∴ ∠x = 115^{o}

**13. Fill in the blanks:**

**(i) If two angles are complementary, then the sum of their measures is _______.**

**Solution: –**

If two angles are complementary, then the sum of their measures is 90^{o}.

**(ii) If two angles are supplementary, then the sum of their measures is ______.**

**Solution: –**

If two angles are supplementary, then the sum of their measures is 180^{o}.

**(iii) Two angles forming a linear pair are _______________.**

**Solution: –**

Two angles forming a linear pair are Supplementary.

**(iv) If two adjacent angles are supplementary, they form a ___________.**

**Solution: –**

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**

**_____________.**

**Solution: –**

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 __________.**

**Solution: –**

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 angles.

**14. In the adjoining figure, name the following pairs of angles.**

**(i) Obtuse vertically opposite angles**

**Solution: –**

∠AOD and ∠BOC are obtuse vertically opposite angles in the given figure.

**(ii) Adjacent complementary angles**

**Solution: –**

∠EOA and ∠AOB are adjacent complementary angles in the given figure.

**(iii) Equal supplementary angles**

**Solution: –**

∠EOB and EOD are the equal supplementary angles in the given figure.

**(iv) Unequal supplementary angles**

**Solution: –**

∠EOA and ∠EOC are the unequal supplementary angles in the given figure.

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

**Solution: –**

∠AOB and ∠AOE, ∠AOE and ∠EOD, ∠EOD and ∠COD are the adjacent angles that do not form a linear pair in the given figure.

## Exercise 5.2

**1. State the property that is used in each of the following statements?**

**(i) If a ****∥**** b, then ****∠****1 = ****∠****5.**

**Solution: –**

Corresponding angles property is used in the above statement.

**(ii) If ****∠****4 = ****∠****6, then a ****∥**** b.**

**Solution: –**

Alternate interior angles property is used in the above statement.

**(iii) If ****∠****4 + ****∠****5 = 180**^{o}**, then a ****∥**** b.**

**Solution: –**

Interior angles on the same side of transversal are supplementary.

**2. In the adjoining figure, identify**

**(i) The pairs of corresponding angles.**

**Solution: –**

By observing the figure, the pairs of corresponding angle are,

∠1 and ∠5, ∠4 and ∠8, ∠2 and ∠6, ∠3 and ∠7

**(ii) The pairs of alternate interior angles.**

**Solution: –**

By observing the figure, the pairs of alternate interior angle are,

∠2 and ∠8, ∠3 and ∠5

**(iii) The pairs of interior angles on the same side of the transversal.**

**Solution: –**

By observing the figure, the pairs of interior angles on the same side of the transversal are ∠2 and ∠5, ∠3 and ∠8

**(iv) The vertically opposite angles.**

**Solution: –**

By observing the figure, the vertically opposite angles are,

∠1 and ∠3, ∠5 and ∠7, ∠2 and ∠4, ∠6 and ∠8

**3. In the adjoining figure, p ∥ q. Find the unknown angles.**

**Solution: –**

By observing the figure,

∠d = ∠125^{o} … [∵ corresponding angles]

We know that, Linear pair is the sum of adjacent angles is 180^{o}

Then,

= ∠e + 125^{o} = 180^{o} … [Linear pair]

= ∠e = 180^{o} – 125^{o}

= ∠e = 55^{o}

From the rule of vertically opposite angles,

∠f = ∠e = 55^{o}

∠b = ∠d = 125^{o}

By the property of corresponding angles,

∠c = ∠f = 55^{o}

∠a = ∠e = 55^{o}

**4. Find the value of x in each of the following figures if l ****∥**** m.**

**(i)**

**Solution: –**

Let us assume other angle on the line m be ∠y,

Then,

By the property of corresponding angles,

∠y = 110^{o}

We know that Linear pair is the sum of adjacent angles is 180^{o}

Then,

= ∠x + ∠y = 180^{o}

= ∠x + 110^{o} = 180^{o}

= ∠x = 180^{o} – 110^{o}

= ∠x = 70^{o}

**(ii)**

**Solution: –**

By the property of corresponding angles,

∠x = 100^{o}

**5. In the given figure, the arms of two angles are parallel.**

**If ****∠****ABC = 70**^{o}**, then find**

**(i) ****∠****DGC**

**(ii) ****∠****DEF**

**Solution: –**

(i) Let us consider that AB ∥ DG

BC is the transversal line intersecting AB and DG

By the property of corresponding angles,

∠DGC = ∠ABC

Then,

∠DGC = 70^{o}

(ii) Let us consider that BC ∥ EF

DE is the transversal line intersecting BC and EF

By the property of corresponding angles,

∠DEF = ∠DGC

Then,

∠DEF = 70^{o}

**6. In the given figures below, decide whether l is parallel to m.**

**(i)**

**Solution: –**

Let us consider the two lines l and m,

n is the transversal line intersecting l and m.

We know that the sum of interior angles on the same side of transversal is 180^{o}.

Then,

= 126^{o} + 44^{o}

= 170^{o}

But, the sum of interior angles on the same side of transversal is not equal to 180^{o}.

So, line l is not parallel to line m.

**(ii)**

**Solution: –**

Let us assume ∠x be the vertically opposite angle formed due to the intersection of the straight line l and transversal n,

Then, ∠x = 75^{o}

Let us consider the two lines l and m,

n is the transversal line intersecting l and m.

We know that the sum of interior angles on the same side of transversal is 180^{o}.

Then,

= 75^{o} + 75^{o}

= 150^{o}

But, the sum of interior angles on the same side of transversal is not equal to 180^{o}.

So, line l is not parallel to line m.

(iii)

**Solution: –**

Let us assume ∠x be the vertically opposite angle formed due to the intersection of the Straight line l and transversal line n,

Let us consider the two lines l and m,

n is the transversal line intersecting l and m.

We know that the sum of interior angles on the same side of transversal is 180^{o}.

Then,

= 123^{o} + ∠x

= 123^{o} + 57^{o}

= 180^{o}

∴The sum of interior angles on the same side of transversal is equal to 180^{o}.

So, line l is parallel to line m.

**(iv)**

**Solution: –**

Let us assume ∠x be the angle formed due to the intersection of the Straight line l and transversal line n,

We know that Linear pair is the sum of adjacent angles is equal to 180^{o}.

= ∠x + 98^{o} = 180^{o}

= ∠x = 180^{o} – 98^{o}

= ∠x = 82^{o}

Now, we consider ∠x and 72^{o} are the corresponding angles.

For l and m to be parallel to each other, corresponding angles should be equal.

But, in the given figure corresponding angles measures 82^{o} and 72^{o} respectively.

∴Line l is not parallel to line m.

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