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Question

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