Chapter 5 Lines and Angles Exercise 5.2

Question 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
Solution:
(i) Given a || b
∴ ∠1 = ∠5 (Pair of corresponding angles)
(ii) Given: ∠4 = ∠6
∴ a || b [If pair of alternate angles are equal, then the lines are parallel]
(iii) Given: ∠4 + ∠5 = 180°
∴ a || b [If sum of interior angles is 180°, then the lines are parallel]

Question 2.
In the given 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.
Solution:
(i) The pair of corresponding angles are ∠1 and ∠5, ∠2 and ∠6, ∠4 and ∠8, ∠3 and ∠7.
(ii) 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 are ∠2 and ∠5, ∠3 and ∠8.

Question 3.
In the given figure, p || q. Find the unknown angles.
Solution:
∠e + 125° = 180° (Linear pair)

∴ ∠e = 180° – 125° = 55°
∠e = ∠f (Vertically opposite angles)
∴ ∠f= 55°
∠a = ∠f= 55° (Alternate interior angles)
∠c = ∠a = 55° (Vertically opposite angles)
∠d = 125° (Corresponding angles)
∠b = ∠d = 125° (Vertically opposite angles)
Thus, ∠a = 55°, ∠b = 125°, ∠c = 55°, ∠d = 125°, ∠e = 55°, ∠f= 55°.

Question 4.
Find the value of x in each of the following figures if l || m Solution:
(i) Let the angle opposite to 110° be y.
∴ y = 110° (Vertically opposite angles)
∠x + ∠y = 180° (Sum of interior angle on the same side of transversal)
∠x + 110° = 180° .
∴ ∠x = 180° – 110° = 70°
Thus x= 70°
(ii) ∠x = 110° (Pair of corresponding angles)

Question 5.
In the given figure, the arms of two angles are parallel. If ∠ABC = 70°, then find
(i) ∠DGC
(ii) ∠DEF

Solution:
Given
AB || DE
BC || EF
∠ABC = 70°
∠DGC = ∠ABC
(i) ∠DGC = 70° (Pair of corresponding angles)
∠DEF = ∠DGC
(ii) ∠DEF = 70° (Pair of corresponding angles)

Question 6.
In the given figure below, decide whether l is parallel to m. Solution:
Sum of interior angles on the same side of transversal
= 126° + 44° = 170° ≠ 180°
∴ l is not parallel to m.

(ii) Let angle opposite to 75° be x.
x = 75° [Vertically opposite angles]
∴ Sum of interior angles on the same side of transversal
= x + 75° = 75° + 75°
= 150° ≠ 180°
∴ l is not parallel to m.

(iii) Let the angle opposite to 57° be y.
∴ ∠y = 57° (Vertically opposite angles)
∴ Sum of interior angles on the same side of transversal
= 57° + 123° = 180°
∴ l is parallel to m.

(iv) Let angle opposite to 72° be z.
∴ z = 70° (Vertically opposite angle)
Sum of interior angles on the same side of transversal
= z + 98° = 72° + 98°
= 170° ≠ 180°
∴ l is not parallel to m.