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, l||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`