NCERT Solutions for Class 7 Maths Chapter 5 Lines and Angles

 

NCERT Solutions for Class 7 Maths Chapter 5 Lines and Angles

Exercise 5.1 Page: 101

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

(i)

Solution:-

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

The given angle is 20o

Let the measure of its complement be xo.

Then,

= x + 20o = 90o

= x = 90o – 20o

= x = 70o

Hence, the complement of the given angle measures 70o.

(ii)

Solution:-

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

The given angle is 63o

Let the measure of its complement be xo.

Then,

= x + 63o = 90o

= x = 90o – 63o

= x = 27o

Hence, the complement of the given angle measures 27o.

(iii)

Solution:-

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

The given angle is 57o

Let the measure of its complement be xo.

Then,

= x + 57o = 90o

= x = 90o – 57o

= x = 33o

Hence, the complement of the given angle measures 33o.

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 180o.

The given angle is 105o

Let the measure of its supplement be xo.

Then,

= x + 105o = 180o

= x = 180o – 105o

= x = 75o

Hence, the supplement of the given angle measures 75o.

(ii)

Solution:-

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

The given angle is 87o

Let the measure of its supplement be xo.

Then,

= x + 87o = 180o

= x = 180o – 87o

= x = 93o

Hence, the supplement of the given angle measures 93o.

(iii)

Solution:-

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

The given angle is 154o

Let the measure of its supplement be xo.

Then,

= x + 154o = 180o

= x = 180o – 154o

= x = 26o

Hence, the supplement of the given angle measures 93o.

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

(i) 65o, 115o

Solution:-

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

Then,

= 65o + 115o

= 180o

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

∴ These angles are supplementary angles.

(ii) 63o, 27o

Solution:-

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

Then,

= 63o + 27o

= 90o

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

∴ These angles are complementary angles.

(iii) 112o, 68o

Solution:-

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

Then,

= 112o + 68o

= 180o

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

∴ These angles are supplementary angles.

(iv) 130o, 50o

Solution:-

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

Then,

= 130o + 50o

= 180o

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

∴ These angles are supplementary angles.

(v) 45o, 45o

Solution:-

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

Then,

= 45o + 45o

= 90o

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

∴ These angles are complementary angles.

(vi) 80o, 10o

Solution:-

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

Then,

= 80o + 10o

= 90o

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

∴ These angles are complementary angles.

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

Solution:-

Let the measure of the required angle be xo.

We know that the sum of measures of complementary angle pair is 90o.

Then,

= x + x = 90o

= 2x = 90o

= x = 90/2

= x = 45o

Hence, the required angle measure is 45o.

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

Solution:-

Let the measure of the required angle be xo.

We know that the sum of measures of supplementary angle pair is 180o.

Then,

= x + x = 180o

= 2x = 180o

= x = 180/2

= x = 90o

Hence, the required angle measure is 90o.

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, which means less than 90o, then they cannot be supplementary because their sum will always be less than 90o.

(ii). Obtuse?

Solution:-

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

(iii). Right?

Solution:-

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

∴ 90o + 90o = 180

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

Solution:-

Let us assume the complementary angles be p and q,

We know that the sum of measures of complementary angle pair is 90o.

Then,

= p + q = 90o

It is given in the question that p > 45o

Adding q on both sides,

= p + q > 45o + q

= 90o > 45o + q

= 90o – 45o > q

= q < 45o

Hence, its complementary angle is less than 45o.

9. 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 90o.

(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 180o.

(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.

10. 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.

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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 = 180o, then a ∥ 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,

∠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 angles 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 = ∠125o … [∵ corresponding angles]

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

Then,

= ∠e + 125o = 180o … [Linear pair]

= ∠e = 180o – 125o

= ∠e = 55o

From the rule of vertically opposite angles,

∠f = ∠e = 55o

∠b = ∠d = 125o

By the property of corresponding angles,

∠c = ∠f = 55o

∠a = ∠e = 55o

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

(i)

Solution:-

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

Then,

By the property of corresponding angles,

∠y = 110o

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

Then,

= ∠x + ∠y = 180o

= ∠x + 110o = 180o

= ∠x = 180o – 110o

= ∠x = 70o

(ii)

Solution:-

By the property of corresponding angles,

∠x = 100o

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

If ∠ABC = 70o, then find

(i) ∠DGC

(ii) ∠DEF

Solution:-

(i) Let us consider AB ∥ DG.

BC is the transversal line intersecting AB and DG.

By the property of corresponding angles

∠DGC = ∠ABC

Then,

∠DGC = 70o

(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 = 70o

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 180o.

Then,

= 126o + 44o

= 170o

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

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 = 75o

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 180o.

Then,

= 75o + 75o

= 150o

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

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 the transversal is 180o.

Then,

= 123o + ∠x

= 123o + 57o

= 180o

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

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 the Linear pair is the sum of adjacent angles equal to 180o.

= ∠x + 98o = 180o

= ∠x = 180o – 98o

= ∠x = 82o

Now, we consider ∠x and 72o 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 82o and 72o, respectively.

∴ Line l is not parallel to line m.