Exercise 5.2 Page: 110

**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 = 180 ^{o}, then a ∥ b.**

**Solution:-**

Interior angles on the same side of transversal are supplementary.

**2. In the adjoining figure, identify**

**(i) The pairs of corresponding angles.**

**Solution:-**

By observing the figure, the pairs of corresponding angle 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 angle 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 = ∠125^{o} … [∵ corresponding angles]

We know that, Linear pair is the sum of adjacent angles is 180^{o}

Then,

= ∠e + 125^{o} = 180^{o} … [Linear pair]

= ∠e = 180^{o} – 125^{o}

= ∠e = 55^{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 other angle on the line m be ∠y,

Then,

By the property of corresponding angles,

∠y = 110^{o}

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

Then,

= ∠x + ∠y = 180^{o}

= ∠x + 110^{o} = 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**

**Solution:-**

(i) Let us consider that AB ∥ DG

BC is the transversal line intersecting AB and DG

By the property of corresponding angles,

∠DGC = ∠ABC

Then,

∠DGC = 70^{o}

(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 = 70^{o}

**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 transversal is 180^{o}.

Then,

= 126^{o} + 44^{o}

= 170^{o}

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

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 = 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}.

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 transversal is 180^{o}.

Then,

= 123^{o} + ∠x

= 123^{o} + 57^{o}

= 180^{o}

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

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

= ∠x + 98^{o} = 180^{o}

= ∠x = 180^{o} – 98^{o}

= ∠x = 82^{o}

Now, we consider ∠x and 72^{o} 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 measures 82^{o} and 72^{o} respectively.

∴Line l is not parallel to line m.