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

(i) If a ∥ b, then ∠1 = ∠5.

Solution:

The property of corresponding angles is used in the above statement.

(ii) If ∠4 = ∠6, then a ∥ b.

Solution:

The property of alternate interior angles is used in the above statement.

(iii) If ∠4 + ∠5 = 180o, then a ∥ b.

Solution:

The property of interior angles on the same side of the transversal is supplementary.

2. In the adjoining figure, identify:

(i) The pairs of corresponding angles.

Solution:

By observing the given figure, the pairs of 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 given 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 given 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 given 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 given figure,

We know that, Linear pair is the sum of adjacent angles i.e 180^o

Then,

 ∠e + 125^o = 180^o  [Linear pair]

 ∠e = 180^o – 125^o

 ∠e = 55^o

Now,

By the property of corresponding angles,

 ∠d = ∠125^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 that the other angle on the line m be ∠b,

Then,

By the property of corresponding angles,

∠b = 110^o

We know that Linear pair is the sum of adjacent angles i.e 180^o

Now,

 ∠x + ∠b = 180^o

 ∠x + 110o = 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

(i) ∠DGC

Solution:

Given, lines A and D are parallel to each other.

 Let us consider that AB ∥ DG

BC is the transversal line that intersects AB and DG

Now,

By the property of corresponding angles,

∠DGC = ∠ABC

Then,

∠DGC = 70^o

(ii) ∠DEF

Solution:

Given, lines A and D are parallel to each other.

Let us consider that BC ∥ EF

DE is the transversal line that intersects BC and EF

Now,

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:

Given, two lines l and m,

L ll m

n is the transversal line that intersects 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, here the sum of interior angles on the same side of transversal is not equal to 180^o.

Hence, line l is not parallel to line m.

(ii)

Solution:

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

Then, ∠a = 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.

Hence, line l is not parallel to line m.

(iii)

Solution:

Let us assume ∠a be the vertically opposite angle that 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 + ∠a

= 123^o + 57^o

= 180^o

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

Hence, line l is parallel to line m.

(iv)

Solution:

Let us assume ∠a be the angle that 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 is equal to 180^o.

∠a + 98^o = 180^o

∠a = 180^o – 98^o

∠a = 82^o

Now, we consider ∠a and 72^o are the corresponding angles.

The corresponding angles should be equal to l and m to become parallel to each other.

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

Hence, line l is not parallel to line m.