# Parallel Lines and Transversal

Question: In the given figure, p || q and l is a transversal. Find the values of x and y.

Solution: 6x + y = x + 5y (Corresponding angles are equal)
Or, 6x – x = 5y – y
Or, 5x = 4y
Or, x = (4y)/(5)

Now, 4x + 6x + y = 180° (linear angles are supplementary)
Or, 10x + y = 180°
Or, (40y)/(5) + y = 180°
Or, (45y)/(5) = 180°
Or, 45y = 180° xx 5 = 900°
Or, y = 20°
Hence, x = (4 xx 20)/(5) = 16°

Question: In the given figure, angle 1 = angle 2, then prove l||m.

Solution: Construction: A transversal intersects two parallel lines on two distinct points.
Given: ∠ 1 = ∠ 2
To prove l||m

Proof: ∠ 1 = ∠ 2 (Given)
∠ 1 = ∠ 3 (Vertically opposite angles are equal)
From above equations, it is clear;
∠ 3 = ∠ 2
Since corresponding angles are equal hence, l||m proved.

Question: In the given figure, ;||m and angle 1 = angle 2, then prove that a||b.

Solution: Construction: Line l||m which are intersected by two transversals a and b.
Given ∠1= ∠2
To prove a||b
Let us name the angle which is vertically opposite to ∠ 3, as ∠ 4.

Proof: ∠ 1 = ∠ 2 (Given)
∠ 1 = ∠ 3 (corresponding angles are equal)
From above equations, it is clear:
∠ 2 = ∠ 3
∠ 3 = ∠ 4 (Vertically opposite angles are equal.
From above equations, it is clear:
∠ 2 = ∠ 4
Since corresponding angles are equal hence, a||b proved.

Question: In the given figure, ∠1 = ∠2 and ∠3 = ∠4, then prove l||m and n||p

Solution: Construction: Lines l and m are intersected by transversals n and p at distinct points.
Given; ∠1 = ∠2 and ∠3 = ∠4
To prove l||m and n||p

Proof: : ∠1 = ∠2 (given)
Since these are corresponding angles and are equal, so l||m is proved.
Now, ∠3 = ∠4 (given)
Since these are alternate interior angles, so n||p is proved.

Question: In the given figure, k||j and m||n, then find the values of x and y.

Solution: Let us name the angle adjacent to 120° as z.
120° + z = 180° (Linear pair of angles is supplementary)
Or, z = 180° - 120° = 60°
∠ x = ∠ z = 60° (Corresponding angles are equal)

Now;
∠ x = ∠ (3y + 6) (Corresponding angles are equal)
Or, 3y + 6 = 60°
Or, 3y = 60° - 6 = 54°
Or, y = 54 ÷ 3 = 18°
Hence, x = 60° and y = 18°

Question: In the following figure, find the pair of parallel lines.

Solution: ∠ MOW ≠ ∠ MPY
So, OW and PY are not parallel
∠ MOX = 50° + 30° = 80°
∠ MOZ = 52° + 28° = 80°
So, ∠ MOX = ∠ MOZ

Since corresponding angles are equal, so OX||OZ
Hence, OX||PZ

Question: In the following figure, a transversal is intersecting two lines at distinct points. Prove l||m.

Solution: ∠ 113° + 67° = 180°
Since internal angles on the same side of transversal are supplementary,
Hence, l||m proved.

Question: In the given figure, a transversal is intersecting two parallel lines at distinct points. Find the value of x.

Solution: 23x – 5 = 21x + 5 (Corresponding angles are equal)
Or, 23x = 21x + 10
Or, 23x – 21x = 10
Or, 2x = 10
Or, x = 5

Question: If u and v are parallel lines, find the value of x.

Solution: Since corresponding angles are equal
Hence, x = 53⁰