Chapter-5 Lines and Angles

(i) Explanation

As we know that, when the summation of two angles is 90o then, the angles are known as complementary angles.

Here, the measurement of one of the angles is given, which is 20o

Now, let’s suppose that the measure of another angle as xo.

So, according to the condition given in the question er can write 

= x + 20o = 90o

= x = 90o – 20o

= x = 70o

Therefore, the complement angle of 20o is 70o.

(ii)

As we know that, when the summation of two angles is 90o then, the angles are known as complementary angles.

Here, the measurement of one of the angles is given, which is 63o

Now, let’s suppose that the measure of another angle as xo.

So, according to the condition given in the question er can write 

= x + 63o = 90o

= x = 90o – 63o

= x = 27o

Therefore, the complement angle of 63o is 27o.

(iii)

As we know that, when the summation of two angles is 90o then the angles are known as complementary angles.

Here, the measurement of one of the angles is given, which is 57o

Now, let’s suppose that the measure of another angle as xo.

So, according to the condition given in the question er can write 

= x + 57o = 90o

= x = 90o – 57o

= x = 33o

Therefore, the complement of 57o is 33o.

(i) Explanation

As we know that, when the summation of two angles is 180o then, the angles are known as supplementary angles.

Here, the measurement of one of the angles is given, which is 105o

Now, let’s suppose that the measure of another angle as xo.

So, according to the condition given in the question er can write 

= x + 105o = 180o

= x = 180o – 105o

= x = 75o

Therefore, the complement angle of 105o is 75o.

(ii) Explanation

As we know that, when the summation of two angles is 180o then, the angles are known as supplementary angles.

Here, the measurement of one of the angles is given, which is 87o

Now, let’s suppose that the measure of another angle as xo.

So, according to the condition given in the question er can write 

= x + 87o = 180o

= x = 180o – 87o

= x = 93o

Therefore, the complement of 87o is 93o.

(iii) Explanation

As we know that, when the summation of two angles is 180o then, the angles are known as supplementary angles.

Here, the measurement of one of the angles is given, which is 154o

Now, let’s suppose that the measure of another angle as xo.

So, according to the condition given in the question er can write 

= x + 154o = 180o

= x = 180o – 154o

= x = 26o

Therefore, the complement of 154o is 26o.

(i) Explanation

As we know that, when the summation of two angles is 90o then the angles are known as complementary angles but when the summation of two angles is 180o then, the angles are known as supplementary angles.

Here, the measurement of one of the angles is given as 65o and the measurement of second angle as 115o.

So, to find whether they are complementary angles or supplementary angles we have to add them. 

= 65o + 115o

= 180o

Since, the summation of the angles is 180o they are supplementary angles.

(ii)

As we know that, when the summation of two angles is 90o then, the angles are known as complementary angles but when the summation of two angles is 180o then, the angles are known as supplementary angles.

Here, the measurement of one of the angles is given as 63o and the measurement of second angle as 27o.

So, to find whether they are complementary angles or supplementary angles we have to add them. 

= 63o + 27o

= 90o

Since, the summation of the angles is 90o they are complementary angles.

(iii)

As we know that, when the summation of two angles is 90o then, the angles are known as complementary angles but when the summation of two angles is 180o then, the angles are known as supplementary angles.

Here, the measurement of one of the angles is given as 112o and the measurement of second angle as 68o.

So, to find whether they are complementary angles or supplementary angles we have to add them. 

= 112o + 68o

= 180o

Since, the summation of the angles is 180o they are supplementary angles.

(iv)

As we know that, when the summation of two angles is 90o then, the angles are known as complementary angles but when the summation of two angles is 180o then, the angles are known as supplementary angles.

Here, the measurement of one of the angles is given as 130o and the measurement of second angle as 50o.

So, to find whether they are complementary angles or supplementary angles we have to add them. 

= 130o + 50o

= 180o

Since, the summation of the angles is 180o they are supplementary angles.

(v)

As we know that, when the summation of two angles is 90o then, the angles are known as complementary angles but when the summation of two angles is 180o then, the angles are known as supplementary angles.

Here, the measurement of one of the angles is given as 45o and the measurement of second angle as 45o.

So, to find whether they are complementary angles or supplementary angles we have to add them. 

= 45o + 45o

= 90o

Since, the summation of the angles is 90o they are complementary angles.

(vi)

As we know that, when the summation of two angles is 90o then, the angles are known as complementary angles but when the summation of two angles is 180o then, the angles are known as supplementary angles.

Here, the measurement of one of the angles is given as 80o and the measurement of second angle as 10o.

So, to find whether they are complementary angles or supplementary angles we have to add them. 

= 80o + 10o

= 90o

Since, the summation of the angles is 90o they are complementary angles.

Explanation

As we know that, when the summation of two angles is 90o then, the angles are known as complementary angles.

Here it is given that both the angle should be equal.

So, let’s suppose that the measure of both the angles as xo.

Now, according to the condition given in the question er can write 

= x + x = 90o

= 2x = 90o

= x = 90o/2

= x = 45o

Therefore, the measurement of the angle is 45o.

Explanation

As we know that, when the summation of two angles is 180o then, the angles are known as supplementary angles.

Here it is given that both the angle should be equal.

So, let’s suppose that the measure of both the angles as xo.

Now, according to the condition given in the question er can write 

= x + x = 180o

= 2x = 180o

= x = 180o/2

= x = 90o

Therefore, the measurement of the angle is 90o.

Explanation

As we know that, when the summation of two angles is 180o then, the angles are known as supplementary angles.

From the figure we can see that

∠1 and ∠2 are supplementary angles. 

Now, when we decrease ∠1, the summation of ∠1 and ∠2 will fall down below 180 and then both the angles would not remain supplementary.

So, to maintain the supplementary angle we need to increase the measurement of angle ∠2 by the same value.

Explanation

(i) supplementary angles.

Now if both the angles are acute angles, which means less than 90o, then the sum of their measurement will be always less than 180o.

No, two acute angles will not be supplementary angles. 

(ii)

As we know that, when the summation of two angles is 180o then, the angles are known as supplementary angles.

Now if both the angles are obtuse angles, which means more than 90o, then the sum of their measurement will be always more than 180o.

No, two obtuse angles will not be supplementary angles.

(iii)

As we know that, when the summation of two angles is 180o then the angles are known as supplementary angles.

Now if both the angles are right angles, which means both the angles will measure 90o, then the sum of their measurement will be equal to 180o.

90o + 90o = 180

yes, two right angles will be supplementary angles. 

Explanation

As we know that, when the summation of two angles is 90o then, the angles are known as complementary angles.

Let’s suppose that the complementary angles are x and y.

= x + y = 90o

But in the question, it is given that one angle is greater than 45 o.

So, suppose that x > 45 o.

Now add y on both the side of above equation,

= x + y > 45o + y

= 90o > 45o + y

= 90o – 45o > y

= y < 45o

Therefore, the measurement of the complementary angle is less than 45o.

Explanation

(i)

Here by looking at the figure, we can say that,

The vertex of ∠1 and ∠2 is same, i.e., O, OC is the common side.

And their non-common sides are OA and OE, are on both the side of the common side.

Yes, ∠1 is adjacent to ∠2.

(ii)

Here by looking at the figure, we can say that,

The vertex of ∠AOC and ∠AOE is same, i.e., O, OA is the common side.

But their non-common sides are not on the both side of common side.

No, ∠AOC is not adjacent to ∠AOE

(iii)

Here by looking at the figure, we can say that,

The vertex of ∠COE and ∠EOD is same, i.e., O, OE is the common side.

And their non-common sides are OC and OD, are on both the side of the common side.

Yes, ∠COE and ∠EOD form a linear pair.

(iv)

Here by looking at the figure, we can say that,

The vertex of ∠BOD and ∠DOA is same, i.e., O, OD is the common side.

And their non-common sides are OA and OB, are on both the side of the common side.

Fron the figure we can say that their summation is also 180o.

Yes, ∠BOD and ∠DOA angles.

(v)

Here by looking at the figure, we can say that,

By the intersection of line AB and CD, ∠1 and ∠2 is formed.

Line AB and CD are two straight lines.

Yes, ∠1 is vertically opposite to ∠2.

(vi)

Here by looking at the figure, we can say that,

By the intersection of line AB and CD, ∠5 and ∠COB is formed.

Line AB and CD are two straight lines. 

So, ∠COB is vertically opposite to ∠5.

Explanation

(i)

By the intersection of two straight lines vertically opposite angle is formed.

Here, by looking at the figure, we can say that,

There is total two vertically opposite angles ∠1 and ∠4, ∠5 and ∠2 + ∠3.

(ii)

The angles which have common vertex and also have non-common side opposite to each other are known as linear pairs.

Here, by looking at the figure, we can say that,

∠1 and ∠5, ∠5 and ∠4 are linear pairs.

Explanation

The angles which have common vertex and also have non-common side opposite to common side are known as adjacent angles.

Here, from the figure we can say that the vertex of ∠1 and ∠2 are not same.

So, ∠1 and ∠2 are not adjacent angles.

Explanation

(i)

Here, from the figure we can say that ∠x + ∠y = 180o and 55o + ∠y = 180o

So, by solving the above equation,

= 55o + ∠y = 180o

= ∠y = 180o – 55o

= ∠y = 125o

Now, ∠y = ∠z because they are vertically opposite angles 

∴ ∠z = 125o

Now, ∠x + ∠y = 180o

=∠x + 125o = 180o

= ∠x = 180o – 125o

= ∠x = 55o

(ii)

Here, from the figure we can say that

∠z and 40o are vertically opposite angles, so ∠z = 40o.

∠y and ∠z are linear pairs, so ∠y + ∠z = 180o.

= ∠y + 40o = 180o

= ∠y = 180o – 40o

= ∠y = 140o

Now, we know that, on a straight line the summation of all the angles is 180o.

40 + ∠x + 25 = 180o 

65 + ∠x = 180o

∠x = 180o – 65

∴ ∠x = 115o

Explanation

(i)

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

Complementary angles are such angles, whose summation always remains 90o.

(ii)

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

Supplementary angles are such angles, whose summation always remains 180o.

(iii)

Two angles forming a linear pair are supplementary.

The angles which have common vertex and also have non-common side opposite to each other are known as linear pairs. Non-common side are opposite to each other means they will form a straight line and according to the condition of straight line, the summation of all the angles on the straight line are always 180o.  which is the same condition for the supplementary angles.

(iv)

If two adjacent angles are supplementary, they form a linear

The angles which have common vertex and also have non-common side opposite to each other are known as linear pairs. Non-common side are opposite to each other means they will form a straight line and according to the condition of straight line, the summation of all the angles on the straight line are always 180o.  which is the same condition for the supplementary angles.

(v)

If two lines intersect at a point, then the vertically opposite angles are always equal.

When two straight lines are intersecting at a point, according to the geometry the angles formed opposite to each other are know as vertically opposite angles and the measurement always remain equal to each other.

(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 angles.

We know that acute angles are those angles whose measurement are less than 90o.  

Now, if one pair is acute angles which means their summation will be less than 180o.

We know that, when two straight lines intersect their summation of all the angles is 360o

So, if the summation of two angle is less than 180o then the summation of other two angles should be more than 180o.

This condition is possible only if the other two angles are obtuse angles

Explanation

(i)

When two straight lines intersect at a point than vertically opposite angles are formed.

From the figure, we can say that ∠AOD and ∠BOC are obtuse vertically opposite angles.

(ii)

Complementary angles are those angles whose sum is equal to 90o.

From the figure, we can say that ∠EOA and ∠AOB are adjacent complementary angles.

(iii)

Supplementary angles are those angles whose sum is equal to 180o

From the figure, we can say that ∠EOB and EOD are the equal supplementary angles because the addition is 180o and both the angles measures same.

(iv)

Supplementary angles are those angles whose sum is equal to 180o

From the figure, we can say that ∠AOB and ∠BOC are the unequal supplementary angles.

(v)

Supplementary angles are those angles whose sum is equal to 180o

From the figure, we can say that ∠AOB and ∠BOC are the unequal supplementary angles.

Explanation

(i)Solution:-

From the figure we can say that corresponding angles property is used.

(ii)Solution:-

From the figure we can say that corresponding angles property is used.

(iii)Solution:-

From the figure we can say that alternate interior angles property is used.

Solution:-

From the geometry of the parallel line, it can be said that interior angles on the same side of the line are supplementary.

Explanation

(i)Solution:-

From the figure, we can say that corresponding angles are as follows,

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

(ii)Solution:-

From the figure, we can say that the pairs of alternate interior angles are,

∠3 and ∠5, ∠2 and ∠8.

(iii) Solution:-

From the figure, we can say that the pairs of interior angles on the same side of the transversal are, ∠3 and ∠8, ∠2 and ∠5.

(iv)Solution:-

From the figure, we can say that, the vertically opposite angles are,

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

Explanation

Solution:-

From the figure, we can say that ∠d and ∠125o  are corresponding angles so,

∠d = ∠125o 

∠e and 125o are Linear pair. So, the sum is 180o

= ∠e + 125o = 180o 

= ∠e = 180o – 125o

= ∠e = 55o

From the figure, we can say that ∠f and ∠e or ∠b and ∠d are vertically opposite angles.

∠f = ∠e = 55o

∠b = ∠d = 125o

From the geometry of the figure, we can say that

∠c = ∠f = 55o

∠a = ∠e = 55o

Explanation

Solution:-

Here from the figure, we can say that lines l and m are parallel to each other.

Now, let’s suppose that the other angle on the line m be ∠y.

So, from the property of line we can say that the sum of the angles on the line is 180o.

Now, from the property of corresponding angles,

∠y = 110o

= ∠x + ∠y = 180o

= ∠x + 110o = 180o

= ∠x = 180o – 110o

= ∠x = 70o

Explanation

Solution:-

(i) Here, let’s suppose that line AB ∥ DG.

From the figure we can say that BC is the transverse line crossing AB and DG.

By applying the property of corresponding angles

∠DGC = ∠ABC

So, ∠DGC = 70o

(ii) Here, let’s suppose that line BC ∥ EF.

From the figure we can say that DE is the transverse line crossing BC and EF.

By applying the property of corresponding angles

∠DEF = ∠DGC

So, ∠DEF = 70o