Question

The sides DC and EB of a cyclic quadrilateral are produced to meet at F, the sides DE and CB are produced to meet at A. If ∠BED = 98° and ∠DFE = 42°, find:

1. ∠BAE 
2. ∠ADC

Answer 1

Given:

∠BED = 98∘

∠DFE = 42∘

Find ∠ADC
Consider triangle ΔADE.

∠BED = 98°

∠DFE = 42°

∠ADC = 180° − 98° − 42°

∠ADC = 180° − 140° = 40°

∠ADC = 40°

Find ∠BAE

∠ADC + ∠ABC = 180° ... [Opposite angles of a cyclic quadrilateral are supplementary.]

∠ABC = 180° − 40° = 140° 

Now consider triangle ΔABF.

At point F, exterior angle:

∠DFE = 42°

Also angle BAF (exterior) = sum of opposite interior angles:

∠BAF = ∠ABC − ∠FBC

∠FBC = 82°    ...[140° − 58° = 82°]

∠BAE = ∠ABC − ∠BAF

∠BAE = 140° − 82° = 58°

∠BAE = 58°