# NCERT Solutions for Class 7 Maths Chapter 5 Lines and Angles

NCERT Solutions for Class 7 Maths Chapter 5 Lines and Angles are the best study materials for those students who are finding difficulties in solving problems.

## Ex -5.1 (Lines and Angles)

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

**(i)**

**Solution:-**

Two angles are 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}Â = 90^{o}

= x = 90^{o}Â â€“ 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}Â = 90^{o}

= x = 90^{o}Â â€“ 63^{o}

= x = 27^{o}

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

**(iii)**

**Solution:-**

Two angles are 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}Â = 90^{o}

= x = 90^{o}Â â€“ 57^{o}

= x = 33^{o}

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

**NCERT Solutions for Class 7 Maths Chapter 4 Simple Equations**

**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}Â = 180^{o}

= x = 180^{o}Â â€“ 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}Â = 180^{o}

= x = 180^{o}Â â€“ 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}Â = 180^{o}

= x = 180^{o}Â â€“ 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}Â + 115^{o}

= 180^{o}

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

âˆ´ 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}Â + 27^{o}

= 90^{o}

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

âˆ´ 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}Â + 68^{o}

= 180^{o}

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

âˆ´ These angles are supplementary angles.

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

**Solution:-**

Then,

= 130^{o}Â + 50^{o}

= 180^{o}

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

âˆ´ These angles are supplementary angles.

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

**Solution:-**

Then,

= 45^{o}Â + 45^{o}

= 90^{o}

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

âˆ´ These angles are complementary angles.

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

**Solution:-**

Then,

= 80^{o}Â + 10^{o}

= 90^{o}

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

âˆ´ These angles are complementary angles.

**4. Find the angles which are equal to their complement.**

**Solution:-**

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

We know that the 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 measure is 45^{o}.

**5. Find the angles which are equal to their supplement.**

**Solution:-**

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

We know that the 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 measure is 90^{o}.

**6. In the given figure, âˆ 1 and âˆ 2 are supplementary angles. If âˆ 1 is decreased, what changes should take place in âˆ 2 so that both angles still remain supplementary?**

**Solution:-**

From the question, it is given that

âˆ 1 and âˆ 2 are supplementary angles.

If âˆ 1 is decreased, then âˆ 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, which means less than 90^{o}, then they cannot be supplementary because their sum will always be less than 90^{o}.

**(ii). Obtuse?**

**Solution:-**

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

**(iii). Right?**

**Solution:-**

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

âˆ´ 90^{oÂ }+ 90^{o}Â = 180

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

**Solution:-**

Let us assume the complementary angles be p and q,

We know that the 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 sides,

= p + q > 45^{oÂ }+ q

= 90^{o}Â > 45^{oÂ }+ q

= 90^{o}Â â€“ 45^{o}Â > q

= q < 45^{o}

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

**9. In the adjoining figure:**

**(i) Is âˆ 1 adjacent to âˆ 2?**

**Solution:-**

By observing the figure, we came to conclude that,

Yes, as âˆ 1 and âˆ 2 have a common vertex, i.e., O and a common arm, OC.

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

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

**Solution:-**

By observing the figure, we came to conclude that,

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

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

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

**Solution:-**

By observing the figure, we came to conclude that,

Yes, as âˆ COE and âˆ EOD have a common vertex, i.e. O and a common arm OE.

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

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

**Solution:-**

By observing the figure, we came to conclude that,

Yes, as âˆ BOD and âˆ DOA have 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 âˆ 1 vertically opposite to âˆ 4?**

**Solution:-**

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

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

**Solution:-**

âˆ COB is the vertically opposite angle of âˆ 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

âˆ 1 and âˆ 4, âˆ 5 and âˆ 2 + âˆ 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,

âˆ 1 and âˆ 5, âˆ 5 and âˆ 4, as these have a common vertex and non-common arms opposite each other.

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

**Solution:-**

âˆ 1 and âˆ 2 are not adjacent angles because they are not lying on the same vertex.

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

**(i)**

**Solution:-**

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

âˆ x + âˆ y = 180^{o}Â â€¦ [âˆµ linear pair]

= 55^{o}Â + âˆ y = 180^{o}

= âˆ y = 180^{o}Â â€“ 55^{o}

= âˆ y = 125^{o}

Then, âˆ y = âˆ z â€¦ [âˆµ vertically opposite angles]

âˆ´ âˆ z = 125^{o}

**(ii)**

**Solution:-**

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

âˆ y + âˆ z = 180^{o}Â â€¦ [âˆµ linear pair]

= âˆ y + 40^{o}Â = 180^{o}

= âˆ y = 180^{o}Â â€“ 40^{o}

= âˆ y = 140^{o}

Then, 40 + âˆ x + 25 = 180^{o}Â â€¦ [âˆµangles on straight line]

65 + âˆ x = 180^{o}

âˆ x = 180^{o}Â â€“ 65

âˆ´ âˆ 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:-**

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

**(ii) Adjacent complementary angles**

**Solution:-**

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

**(iii) Equal supplementary angles**

**Solution:-**

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

**(iv) Unequal supplementary angles**

**Solution:-**

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

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

**Solution:-**

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

## Exercise 5.2 (Lines and Angles)

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

**(i) If a âˆ¥ b, then âˆ 1 = âˆ 5.**

**Solution:-**

Corresponding angles property is used in the above statement.

**(ii) If âˆ 4 = âˆ 6, then a âˆ¥ b.**

**Solution:-**

Alternate interior angles property is used in the above statement.

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

**Solution:-**

Interior angles on the same side of the transversal are supplementary.

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

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

**Solution:-**

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

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

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

**Solution:-**

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

âˆ 2 and âˆ 8, âˆ 3 and âˆ 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 âˆ 2 and âˆ 5, âˆ 3 and âˆ 8

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

**Solution:-**

By observing the figure, the vertically opposite angles are,

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

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

**Solution:-**

By observing the figure,

âˆ d = âˆ 125^{o}Â â€¦ [âˆµ corresponding angles]

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

Then,

= âˆ e + 125^{o}Â = 180^{o}Â â€¦ [Linear pair]

= âˆ e = 180^{o}Â â€“ 125^{o}

= âˆ e = 55^{o}

From the rule of vertically opposite angles,

âˆ f = âˆ e = 55^{o}

âˆ b = âˆ d = 125^{o}

By the property of corresponding angles,

âˆ c = âˆ f = 55^{o}

âˆ a = âˆ e = 55^{o}

**UNDERSTANDING POLLUTION: TYPES, IMPACTS, AND SOLUTIONS FOR A SUSTAINABLE FUTURE**

**4. Find the value of x in each of the following figures if l âˆ¥ m. (Lines and Angles)**

**(i)**

**Solution:-**

Let us assume the other angle on the line m be âˆ y.

Then,

By the property of corresponding angles,

âˆ y = 110^{o}

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

Then,

= âˆ x + âˆ y = 180^{o}

= âˆ x + 110^{o}Â = 180^{o}

= âˆ x = 180^{o}Â â€“ 110^{o}

= âˆ x = 70^{o}

**(ii)**

**Solution:-**

By the property of corresponding angles,

âˆ x = 100^{o}

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

**If âˆ ABC = 70 ^{o}, then find**

**(i) âˆ DGC**

**(ii) âˆ DEF**

**Solution:-**

(i) Let us consider AB âˆ¥ DG.

BC is the transversal line intersecting AB and DG.

By the property of corresponding angles

âˆ DGC = âˆ ABC

Then,

âˆ DGC = 70^{o}

(ii) Let us consider that BC âˆ¥ EF.

DE is the transversal line intersecting BC and EF.

By the property of corresponding angles

âˆ DEF = âˆ DGC

Then,

âˆ 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 the transversal is 180^{o}.

Then,

= 126^{o}Â + 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 âˆ x be the vertically opposite angle formed due to the intersection of the straight line l and transversal n,

Then, âˆ 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 the transversal is 180^{o}.

Then,

= 75^{o}Â + 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 âˆ x is 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 the transversal is 180^{o}.

Then,

= 123^{o}Â + âˆ x

= 123^{o}Â + 57^{o}

= 180^{o}

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

So, line l is parallel to line m.

**(iv)**

**Solution:-**

Let us assume âˆ x is the angle formed due to the intersection of the Straight line l and transversal line n.

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

= âˆ x + 98^{o}Â = 180^{o}

= âˆ x = 180^{o}Â â€“ 98^{o}

= âˆ x = 82^{o}

Now, we consider âˆ x and 72^{o}Â 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 measure 82^{o}Â and 72^{o}, respectively.

âˆ´ Line l is not parallel to line m.