Question
Download Solution PDFSides AB and DC of a cyclic quadrilateral ABCD are produced to meet at E and sides AD and BC are produced to meet at F. If ∠ADC = 78° and ∠BEC = 52°, then the measure of ∠AFB is:
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
∠ADC = 78° and ∠BEC = 52°
Concept:
The sum of a pair of opposite angles is 180° of a cyclic quadrilateral.
Calculation:
Now, In quadrilateral ABCD,
∠ADC + ∠ABC = 180° (Sum of opposite angles are 180°)
⇒ ∠ABC = 180° - 78° = 102°
∠EBC = 180° - 102° = 78° [Linear Pair]
In ΔBEC,
∠BEC + ∠EBC + ∠BCE = 180°
⇒ 52° + 78° + ∠BCE = 180°
⇒ ∠BCE = 180° - 130° = 50°
⇒ ∠BCE = ∠DCF = 50° (Vertically opposite angle)
∠ADE + ∠FDC = 180° [Linear Pair]
⇒ ∠FDC = 180° - ∠ADE = 180° - 78° = 102°
In ΔDCF,
∠FDC + ∠DCF + ∠DFC = 180°
102 + 50 + ∠DFC = 180°
∠DFC = 180° - 152° = 28°
∠DFC = ∠AFB = 28°
∴ The measure of ∠AFB is 28°.
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