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.

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Solution

## 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∘.

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