Question

In Fig. 113 if AC || OF and AB || CE, then
(a) x = 145, y = 223
(c) x = 135, y = 233
(b) x = 223, y = 145
(d) x = 233, y = 135

Answer 1

Construction: Produce FD towards D to the point M
∠DCA + Reflex∠DCA = 360^∘ [Complete angle]
∴∠DCA + (y + 15)^∘ = 360^∘
⇒ ∠DCA = 345^∘ − y^∘
Now,
∠MDC = ∠EDF = 58^∘ [Vertically Opposite angles]
Since, MF || AC
∴ ∠MDC + ∠QPD = 180^∘ [Angles on the same side of a transversal line are supplementary]
⇒ 58^∘ + 345^∘ − y^∘ = 180^∘
⇒ y = 223
∴ ∠DCA = 345^∘ − 223^∘ = 122^∘
Again, ∠BAC + Reflex∠BAC = 360^∘ [Complete angle]
∴∠BAC + (2x + 12)^∘ = 360^∘
⇒ ∠DCA = 348^∘ − (2x)^∘
Since, AB || CD
∴ ∠DCA + ∠DCA = 180^∘ [Angles on the same side of a transversal line are supplementary]
⇒ 348^∘ − (2x)^∘ + 122^∘ = 180^∘
⇒ (2x)^∘ = 290^∘
⇒ x = 145
Hence, the correct answer is option (a).