**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.