Question

Line AB || line CD || line EF and line QP is their transversal. If y : z = 3 : 7 then find the measure of $\angle$x.

Answer 1

Suppose the transversal QP intersects line AB, line CD and line EF at K, L and M respectively.
Since line CD and line QP intersect at L, then
∠DLM = ∠CLK (Vertically opposite angles)
⇒∠DLM = ∠y

Since line CD || line EF and QP is a transversal intersecting them at L and M, then
∠DLM + ∠FML = 180^∘ (Pair of interior angles on the same side of transversal is supplementary)
⇒∠y + ∠z = 180^∘ ....(1)
We have, y : z = 3 : 7. Let y = 3k and z = 7k
Now, from (1), we get
3k + 7k = 180^∘
⇒10k = 180^∘
⇒k = 18^∘
∴ ∠y = 3k = 3 × 18^∘ = 54^∘ and ∠z = 7k = 7 × 18^∘ = 126^∘

Since line AB || line CD and QP is a transversal intersecting them at K and L, then
∠AKL + ∠CLK = 180^∘ (Pair of interior angles on the same side of transversal is supplementary)
⇒∠x + ∠y = 180^∘
⇒∠x + 54^∘ = 180^∘
⇒∠x = 180^∘ − 54^∘ = 126^∘

So, measure of ∠x is 126^∘.