Give that segment AB and CD are parallel, if lines ℓ, m and n intersect at point O. Find the ratio of θ to ∠ODS

Let the line m cut AB and CD at point P and Q respectively

∠ DOQ = x (exterior angle)

Hence, Y + 2x (corresponding angle)

∴ y = x ...(1)

Also . ∠ DOQ = x (vertically opposite angles)

In ∆ OCD, sum of the angles = 180⁰

∴ y + 2y + 2x + x =180°

⇒ 3x + 3y = 180°

⇒ x + y = 60 ...(2)

From (1) and (2)

x = y = 30 = 2y = 60

∴ ∠ ODS = 180 – 60 = 120°

∴ θ = 180 – 3x = 180 – 3(30) = 180 – 90 = 90°.

∴ The required ratio = 90 : 120 = 3 : 4.