Answer :

In Δ FBD

∠ BFD + ∠ FBD + ∠ BDF = 180° (angle sum property)

⇒ ∠ BFD + 50 + 30 = 180°

⇒ ∠ BFD = 180° – 80° = 100°

Also, ∠ CFE = ∠ BFD =100° (∵ ∠ BFD and ∠ CFE are vertically

opp. Angle)

Since, EFD is a line

∠ BFD + ∠ BFE = 180° (sum of angle on a straight line is 180°)

⇒ 100 + ∠ BFE = 180°

⇒ ∠ BFE = 180° – 100° = 80°

Also, ∠ CFD = ∠ BFE = 80° (∵ ∠ BFE and ∠ CFD are vertically opp. Angle)

In Δ FBE

∠ BFE + ∠ FBE + ∠ BEF = 180° (angle sum property)

⇒ ∠ FBE + 80 + 45 = 180°

⇒ ∠FBE = 180° – 125° = 55°

Thus, ∠ DBE = ∠ FBD + ∠FBE

⇒ ∠ DBE = 30 + 55 = 85°

In quad CDBE

∠ DCE + ∠ DBE = 180° (∵ if all four vertices of a quadrilateral are

on circle then opposite angle are supplementary)

⇒ ∠ DCE + 85 = 180°

⇒ ∠ DCE = 180° – 85° = 95°

Also,

∠ CBD = ∠ DEC = 30° (∵ angle in a same segment are equal)

∠ CBE = ∠ CDE = 55° (∵ angle in a same segment are equal)

Thus, ∠ CDB = ∠ CDE + ∠ BDE

⇒ ∠ CDB = 55 + 50 = 105°

Thus, ∠ CEB = ∠ CED + ∠ BED

⇒ ∠ CEB = 30 + 45 = 75°

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