Lines And Angles

(i) We know that,

Sum of measures of complementary angles is 90^o

It is given in the question that,

Angle = 20^o

Therefore,

Complement = 90^o – 20^o

= 70^o

(ii) We know that,

Sum of measures of complementary angles is 90^o

It is given in the question that,

Angle = 63^o

Therefore,

Complement = 90^o – 63^o

= 27^o

(iii) We know that,

Sum of measures of complementary angles is 90^o

It is given in the question that,

Angle = 57^o

Therefore,

Complement = 90^o – 57^o

= 33^o

(i) We know that,

Sum of measures of supplementary angles is 180^o

It is given in the question that,

Angle = 105^o

Therefore,

Supplement = 180^o – 105^o

= 75^o

(ii) We know that,

Sum of measures of supplementary angles is 180^o

It is given in the question that,

Angle = 87^o

Therefore,

Supplement = 180^o – 87^o

= 93^o

(iii) We know that,

Sum of measures of supplementary angles is 180^o

It is given in the question that,

Angle = 154^o

Therefore,

Supplement = 180^o – 154^o

= 26^o

(i) We know that,

Sum of measures of supplementary angles is 180^o

Also,

Sum of measures of complementary angles is 90^o

It is given in the question that,

The two pairs of angles are 115^o and 65^o

Therefore,

Sum of the measures of these angles = 115^o + 65^o

= 180^o

As the sum of these angles is equal to 180^o

Therefore,

These angles are supplementary angles.

(ii) We know that,

Sum of measures of supplementary angles is 180^o

Also,

Sum of measures of complementary angles is 90^o

It is given in the question that,

The two pairs of angles are 63^o and 27^o

Therefore,

Sum of the measures of these angles = 63^o + 27^o

= 90^o

As the sum of these angles is equal to 90^o

Therefore,

These angles are complementary angles

(iii) We know that,

Sum of measures of supplementary angles is 180^o

Also,

Sum of measures of complementary angles is 90^o

It is given in the question that,

The two pairs of angles are 112^o and 68^o

Therefore,

Sum of the measures of these angles = 112^o + 68^o

= 180^o

As the sum of these angles is equal to 180^o

Therefore,

These angles are supplementary angles

(iv) We know that,

Sum of measures of supplementary angles is 180^o

Also,

Sum of measures of complementary angles is 90^o

It is given in the question that,

The two pairs of angles are 130^o and 50^o

Therefore,

Sum of the measures of these angles = 130^o + 50^o

= 180^o

As the sum of these angles is equal to 180^o

Therefore,

These angles are supplementary angles

(v) We know that,

Sum of measures of supplementary angles is 180^o

Also,

Sum of measures of complementary angles is 90^o

It is given in the question that,

The two pairs of angles are 45^o and 45^o

Therefore,

Sum of the measures of these angles = 45^o + 45^o

= 90^o

As the sum of these angles is equal to 90^o

Therefore,

These angles are complementary angles

(vi) We know that,

Sum of measures of supplementary angles is 180^o

Also,

Sum of measures of complementary angles is 90^o

It is given in the question that,

The two pairs of angles are 80^o and 10^o

Therefore,

Sum of the measures of these angles = 80^o + 10^o

= 90^o

As the sum of these angles is equal to 90^o

Therefore,

These angles are complementary angles

We have to find out the angle which is equal to its complement

Let us assume the angle be x

Also,

Complement of this angle is also x

We know that,

Sum of the measures of complementary angles = 90^o

Therefore,

x + x = 90^o

2x = 90^o

x =

x = 45^o

We have to find out the angle which is equal to its supplement

Let us assume the angle be x

Also,

Supplement of this angle is also x

We know that,

Sum of the measures of supplementary angles = 180^o

Therefore,

x + x = 180^o

2x = 180^o

x =

x = 90^o

It is given in the question that,

∠1 and ∠2 are supplementary angles

And if ∠1 is decreased than ∠2 must be increased by the same measure so that the pair of both the angles remain supplementary.

(i) If both the angles are acute then two angles can’t be supplementary because:

Acute angle is always less than 90^o and addition of any two angles less than 90^o can never be equal to 180^o

Hence,

Two acute angles cannot be in a supplementary angle pair

(ii) If both the angles are obtuse then two angles can’t be supplementary because:

Obtuse angle is always greater than 90^o and addition of any two angles greater than 90^o can never be equal to 180^o

Hence,

Two obtuse angles cannot be in a supplementary angle pair

(iii) if both the angles are right angles then the pair of angles must be supplementary because:

Right angle is equal to 90^o and addition of two right angles is equal to 180^o

i.e., 90^o + 90^o = 180^o

Hence,

Two right angles form a supplement angle of pair

Let us assume two angles x and y

Now,

x and y are two angles making a complementary angle pair and x is greater 45^o

Therefore,

x + y = 90^o

y = 90^o – x^o

As x^o > 45^o


- x^o < - 45^o

Add 90^o on both sides to get,

90^o - x^o < 90^o - 45^o
y^o < 90^o - 45^o

y^o < 45^o
Hence,

Angle y will be lesser than 45^o.

(i) Yes, ∠1 is adjacent to ∠2 as:

Both the angles have common vertex O

Also,

Both the angles have a common arm OC

Also,

The non – common arms of both the angles i.e., OA and OC are on either side of the common arm

(ii) No, ∠AOC is not adjacent to ∠AOE as:

Both the angles have common vertex O

Also,

Both the angles have a common arm OA

But,

The non – common arms of both the angles i.e. OC and OE are on the same side of the common arm

(iii) Yes, ∠COE and ∠EOD form a linear pair because:

Both the angles have common vertex O

Also,

Both the angles have a common arm OE

Also,

The non – common arms of both the angles i.e., OC and OD are opposite rays

(iv) Yes, ∠BOD and ∠DOA are supplementary because:

Both the angles have common vertex O

Also,

The non – common arms of both the angles are opposite to each other

(v) Yes, ∠1 and ∠2 are vertically opposite to each other because:

These angles are formed due to the intersection o two straight lines i.e., AB and CD

(vi) ∠COB is the vertically opposite angle of ∠5 because these are formed due to the intersection of two straight lines i.e., AB and CD

(i) Following pairs of vertically opposite angles are as follows:

∠1 and ∠4

Also,

∠5 and ∠2 + ∠3

These all are pairs of vertically opposite angles because these are formed due to the intersection of two straight lines

(ii) Following pairs of linear pairs are as follows:

∠1 and ∠5

Also,

∠5 and ∠4

These all pairs are linear pairs because these all have a common vertex and their non – common arms are opposite to each other

In the above figure,

∠1 and ∠2 are not adjacent because the angles have no common vertex

(i) We have to find the value of x, y and z

We have,

∠x and 55^o are vertically opposite angles

Therefore,

∠x = 55^o

Also,

∠x and ∠y form a linear pair

Therefore,

∠x + ∠y = 180^o (Sum of linear pair angles)

55^o + ∠y = 180^o

∠y = 180^o – 55^o

= 125^o

Also,

∠y and ∠z are vertically opposite angles

Therefore,

∠y = ∠z = 125^o

Hence,

The value of x, y and z is as follows:

∠x = 55^o

∠y = 125^o

And,

∠z = 125^o

(ii) We have to find out the values of x, y and z

We have,

∠z and 40^o are vertically opposite angles

Therefore,

∠z = 40^o

Also,

∠y and ∠z form a linear pair

Therefore,

∠y + ∠z = 180^o (Sum of angles of linear pair)

∠y + 40^o = 180^o

∠y = 180^o – 40^o

= 140^o

Also, we know that:

Sum of angles in a straight line = 180^o

∠x + 40^o + 25^o = 180^o

∠x + 40^o + 25^o = 180^o

∠x + 65^o = 180^o

∠x = 180^o – 65^o

= 115^o

Hence,

The value of ∠x, ∠y and ∠z is as follows:

∠x = 115^o

∠y = 140^o

∠z = 40^o

(i) 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 180^o

(iii) Two angles forming a linear pair are supplementary

(iv) If two adjacent angles are supplementary, they form a linear pair

(v) If two lines intersect at a pint, 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 Obtuse angle

(i) From the given figure,

Obtuse vertically opposite angles are:

∠AOD, ∠BOC

(ii) From the given figure,

Adjacent complementary angles are:

∠EOA, ∠AOB

(iii) From the given figure,

Equal supplementary angles are:

∠EOB, ∠EOD

(iv) From the given figure,

Unequal supplementary angles are:

∠EOA, ∠EOC

(v) From the given figure,

Adjacent angles that do not form a linear pair are:

∠AOB and ∠AOE

∠AOE and ∠EOD

Also,

∠EOD and ∠COD

(i) From the above-given figure, it is clear that,

∠1 = ∠5

This is because a is parallel to b also ∠1 and ∠5 are corresponding angles

Hence,

In these angles are equal due to corresponding angles property

(ii) From the above-given figure, it is clear that,

∠4 = ∠6

This is because of the alternate interior angles property

(iii) We have,

From the above-given figure, it is clear that,

∠4 + ∠5 = 180^o

Because we know that, the interior angles on the same side of the transversal are supplementary

(i) From the given figure, we have

Pairs of corresponding angles are as follows:

∠1 and ∠5

∠2 and ∠6

∠3 and ∠7

∠4 and ∠8

(ii) From the given figure, we have

Pairs of alternate interior angles are as follows:

∠2 and ∠8

∠3 and ∠5

(iii) From the given figure, we have

Pairs of alternate interior angles on the same side of the transversal are as follows:

∠2 and ∠5

∠3 and ∠8

(iv) From the given figure, we have

Pairs of vertically opposite angles are as follows:

∠1 and ∠3

∠2 and ∠4

∠5 and ∠7

∠6 and ∠8

From the above figure, the unknown angles are as follows:

∠d = 125^o (Corresponding angles)

∠e = 180^o – 125^o

= 55^o (Linear pair)

∠f = ∠e = 55^o (Vertically opposite angles)

∠c = ∠f = 55^o (Corresponding angles)

∠a = ∠e = 55^o (Corresponding angles)

∠b = ∠d = 125^o (Vertically opposite angles)

(i) From the above figure, we have

∠y and 110^o are equal as both are corresponding angles

Therefore,

∠y = 110^o (Corresponding angles)

Also,

∠x and ∠y are forming a linear pair and we know that sum of angles of linear pair is equal to 180^o

Therefore,

∠x + ∠y = 180^o (Linear pair)

∠x + 110^o = 180^o

∠x = 180^o – 110^o

∠x = 70^o

Hence, value of ∠x is 70^o

(ii) From the above figure it is clear that,

∠x and 100^o are equal as they are corresponding angles

Therefore,

∠x = 100^o

Hence,

The value of ∠x is 100^o

(i) From the figure, it is clear that AB is parallel to DG and there is a transversal line BC that is intersecting them

Therefore,

∠DGC = ∠ABC (Corresponding angles)

As, it is given in the question that:

∠ABC = 70^o

Therefore,

∠ABC = ∠DGC = 70^o

Hence,

The value of ∠DGC is equal to 70^o

(ii) From the figure, it is clear that BC is parallel to EF and there is a transversal line DE that is intersecting them

Therefore,

∠DEF = ∠DGC (Corresponding angles)

As, from result of (i) it is clear that:

∠DGC = 70^o

Therefore,

∠DGC = ∠DEF = 70^o

Hence,

The value of ∠DEF is equal to 70^o

(i) From the above figure,

There are two angles 126^o and 44^o which are on the same side of the transversal line n

Let us consider two lines l and m

Also,

There is a transversal line n which is intersecting both the lines

Therefore,

Sum of interior angles on the same side of the transversal = 126^o + 44^o

= 170^o

Hence,

It is clear that the sum of interior angles which are on the same side of the transversal is not equal to 180^o

Therefore,

The lines l and m are not parallel to each other

(ii) In the above question we have, Angle x and 75^o are forming a linear pair on the line l

Therefore,

x + 75^o = 180^o (Sum of angles of linear pair)

x = 180^o – 75^o

= 105^o

We have to show that l and m are parallel to each other

For this,

Corresponding angles ∠ABC and ∠x should be equal

But,

∠x = 105^o

And,

∠ABC = 75^o

Therefore,

Lines l and m are not parallel to each other

(iii) In the above question we have, Angle x and 123^o are forming a linear pair on the line m

Therefore,

x + 123^o = 180^o (Sum of angles of linear pair)

x = 180^o – 123^o

= 57^o

We have to show that l and m are parallel to each other

For this,

Corresponding angles ∠ABC and ∠x should be equal

∠x = 57^o

And,

∠ABC = 57^o

Therefore,

Both the angles are equal to each other

Hence,

Lines l and m are parallel to each other.

(iv) In the above question we have, Angle x and 98^o are forming a linear pair on the line l

Therefore,

x + 98^o = 180^o (Sum of angles of linear pair)

x = 180^o – 98^o

= 82^o

We have to show that l and m are parallel to each other.

For this,

Corresponding angles ∠ABC and ∠x should be equal

But,

∠x = 82^o

And,

∠ABC = 72^o

Therefore,

Lines l and m are not parallel to each other.