Question

In Fig. 112, if AB || CD, then x =
(a) 32
(b) 42
(c) 52
(d) 31

Answer 1

Construction: Draw a line PQ parallel to AB which is also parallel to CD
∠CDP + Reflex∠CDP = 360^∘ [Complete angle]
∴∠CDP + 249^∘ = 360^∘
⇒ ∠CDP = 111^∘
Since, PQ || AB
∴ ∠BAP = ∠APQ [Alternate angles]
⇒ ∠BAP = 28^∘
Now, ∠APQ + ∠QPD = ∠APD
⇒ 28^∘ + ∠QPD = (2x + 13)^∘
⇒ ∠QPD = (2x + 13)^∘ − 28^∘
Since, PQ || CD
∴ ∠QPD + ∠CDP = 180^∘ [Angles on the same side of a transversal line are supplementary]
⇒ (2x + 13)^∘ − 28^∘ + 111^∘ = 180^∘
⇒ 2x + 13 − 28 + 111 = 180
⇒ 2x = 84
⇒ x = 42
Hence, the correct answer is option (b).