**Lines and Angles**

**Exercise 5.2**

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

(i) If *a *|| *b*, then ∠1 = ∠5.

(ii) If ∠4 = ∠6, then *a *|| *b*.

(iii) If ∠4 + ∠5 = 180°, then *a *|| *b*.

Answer:

(i) If *a *|| *b*, then ∠1 = ∠5.

Corresponding angles property.

(ii) If ∠4 = ∠6, then *a *|| *b*.

The converse of alternate angles property.

(iii) If ∠4 + ∠5 = 180°, then *a *|| *b*.

Interior angles property.

2). In the adjoining figure, identify

(i) the pairs of corresponding angles.

(ii) the pairs of alternate interior angles.

(iii) the pairs of interior angles on the same side of the transversal.

(iv) the vertically opposite angles.

Answer:

(i) the pairs of corresponding angles.

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

(ii) the pairs of alternate interior angles.

∠4 and ∠6 and ∠3 and ∠5

(iii) the pairs of interior angles on the same side of the transversal.

∠3 and ∠6 and ∠4 and ∠5

(iv) the vertically opposite angles.

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

3). In the adjoining figure, *p *|| *q*. Find the unknown angles.

125^{0} and ∠*e* are forming linear pair

125^{0} + ∠*e *= 180

*∠e *= 180 – 125

*∠e *= 55^{0}

*∠e* and ∠*f *are vertically opposite angles

*∠e* = ∠*f*

But ∠*e *= 55^{0}

*∠f *= 55^{0}

125^{0} and *∠**d* are pair of corresponding angles

∠*d* = 125^{0}

*∠d* and ∠*b *are vertically opposite angles

*∠d* = ∠*b*

But ∠*d *= 125^{0}

*∠b *= 125^{0}

*∠d* and ∠*c* are forming linear pair

∠*d* + ∠*c *= 180

But ∠*d* + = 125

*∠c *= 180 – 125

*∠c *= 55^{0}

*∠a* and ∠*c *are vertically opposite angles

*∠a* = ∠*c*

But ∠*c *= 55^{0}

*∠a *= 55^{0}

4). Find the value of *x *in each of the following figures if *l *|| *m.*

(i) (ii)

Solution:

(i) let *y* be the vertically opposite angle of 110^{0}

*y* = 110^{0 }(vertically opposite angles are congruent)

*l *|| *m *and* t *is transversal

110^{0} + *x*^{0} = 180 (interior angles are supplementary)

*x*^{0} =180 – 110

*x*^{0} =70^{0}

(ii) *l *|| *m *and* a *is transversal

*x* = 100^{0 }(corresponding angles are congruent)

5). In the given figure, the arms of two angles are parallel. If ∠ABC = 70º, then find

(i) ∠DGC

(ii) ∠DEF

*Given:* ∠ABC = 70º

AB ||DE and BC || EF

Since AB ||DE and BC is transversal

∠ABC and ∠DGC are forming a pair of corresponding angles

∠ABC = ∠DGC

But ∠DGC = 70^{0}

∠DGC = 70^{0}

Since BC ||EF and DE is transversal

∠DEF and ∠DGC are forming a pair of corresponding angles

∠DEF = ∠DGC

But ∠DGC = 70^{0}

∠DEF = 70^{0}

6). In the given figures below, decide whether *l *is parallel to *m*.

i).

126^{0} + 44^{0} = 170

126^{0} + 44^{0} ≠180

The Sum of interior angles is not equal to 180.

By interior angles test, *l *is not parallel to *m *

ii).

let *x* be the vertically opposite angle of 75^{0}

*x* = 75^{0 }(vertically opposite angles are congruent)

75^{0} + 75^{0} = 150

75^{0} + 75^{0} ≠180

The Sum of interior angles is not equal to 180

By interior angles test, *l *is not parallel to *m *

iii).

let *x* be the vertically opposite angle of 57^{0}

*x* = 57^{0 }(vertically opposite angles are congruent)

57^{0} + 123^{0} = 180

The Sum of interior angles is equal to 180

By interior angles test, *l *is parallel to *m *

iv).

let *x* be the vertically opposite angle of 72^{0}

*x* = 72^{0 }(vertically opposite angles are congruent)

72^{0} + 98^{0} = 170

75^{0} + 75^{0} ≠180

The Sum of interior angles is not equal to 180

By interior angles test, *l *is not parallel to *m *

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