Question

In the figure, 𝐀𝐁𝐂𝐃 is a cyclic quadrilateral in which 𝐀𝐂 and 𝐁𝐃 are its diagonals. If ∠𝐃𝐁𝐂 =𝟓𝟓°, ∠𝐃𝐂𝐀 =𝟒𝟎°, and ∠𝐀𝐃𝐁 =𝟐𝟓° find ∠𝐁𝐃𝐂.

Answer 1

We know that the angle subtended by arc in the same segment are equal therefore:
∠DCA = ∠DBA = 40°

From figure:
∠ABC = ∠ABD + ∠DBC
∠ABC = 40° + 55° = 95°
Since ABCD is a cyclic quadrilateral:
∠ABC + ∠ADC = 180°
95° + ∠ADC = 180°
therefore, ∠ADC = 85°
∠ADC = ∠ADB + ∠BDC
25 + ∠BDC = 85
∠BDC = 85° - 25° = 60°
∠BDC = 60°