Exercise 5.2

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

a) If a // b , then ∠1 = ∠5
Ans: Corresponding angle property.
b) If ∠4 = ∠6 , then a // b 3 4

Ans: Alternate interior angle property. 5 6
7 8
c) If ∠4 + ∠5 = 180ᵒ , then a // b
Ans: Interior angles on the same side of the transversal are supplementary.

2) In the adjoining figure, identify.

i) The pairs corresponding angles
Ans: ∠1 and ∠5 , ∠2 and ∠6 4 1
∠4 and ∠8 , ∠3 and ∠7
ii) The pairs of alternate interior angles 7 6
Ans: ∠2 and ∠8 , ∠3 and ∠5
iii) The pairs of interior angles on the same side of the transversal

Ans: ∠2 and ∠5 , ∠3 and ∠8
iv) The vertically opposite angles

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

3) In the adjoining figure, p // q. Find the unknown angles.
Ans:

d = 125ᵒ —— corresponding angle

b = 125ᵒ ——- vertically opposite angles 
125ᵒ + ∠e = 180ᵒ
∠e = 180ᵒ − 125ᵒ 
∠e = 55ᵒ 
∠e = ∠f —— vertically opposite angles 
∠f = 55ᵒ
∠e = ∠a ——– corresponding angle
∠a = 55ᵒ
∠a = ∠c ——— vertically opposite angles
∠c = 55ᵒ

4) Find the value of x in each of the following figures if l // m.
Ans:

i) Let us assume other angle on the line in as ∠y
∠y + 110ᵒ = 180ᵒ ——- linear pair
∠y = 180ᵒ – 110ᵒ
∠y = 70ᵒ
∠y = ∠x ——– corresponding angles
∴ ∠x = 70ᵒ
ii) ∠x = 100ᵒ ——— corresponding angles property

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

i) ∠DGC
Ans: Let us consider that AB // DG and BC as its transversal
∠ ABC = ∠DGC ——- corresponding angles
∠DGC = 70ᵒ
ii) ∠DEF

Ans: Consider BC // EF and DE as its transversal
∠DGC = ∠DEF ——— corresponding angles
∠DEF = 70ᵒ

6) In the figure below, decide whether l is parallel to m.

                                (iii)                                                                                                                      (iv)
Ans:

i) n is the transversal to lines m, n. We know that if interior angles on the same side of the transversal are supplementary then the lines are parallel so 126ᵒ + 44ᵒ= 170ᵒ
∴ l and m are not parallel.
ii) Let us assume ∠x to be vertically opposite angle formed on line l.
Such that ∠x = 75ᵒ
We know that if the interior angles on the same side of the transversal are supplementary then the lines are parallel.
So, 75ᵒ + 75ᵒ
= 150ᵒ
Therefore, l and m are not parallel.
iii) n is the transversal to lines l and m. Let us assume ∠x to be vertically opposite angles formed on line l such that ∠x = 57ᵒ
We know that if the interior angles on the same side of the transversal are supplementary then the lines are parallel.
So, 57ᵒ + 123ᵒ
= 180ᵒ
∴ l and m are parallel lines.
iv) Let ∠x be the angle formed by the intersection of the line l and the transversal n.
Such that ∠x + 98ᵒ = 180ᵒ
∠x = 180ᵒ – 98ᵒ
∠x = 82ᵒ
∠x and 72ᵒ are corresponding angles. If the corresponding angles are equal then the two lines are parallel.
Since, ∠x = 82ᵒ and 72ᵒ are not equal, hence line l and m are not parallel.