In the figure below, ∠𝐷𝐸𝐶 + ∠𝐵𝐹𝐶 is equal to ____________ .

Key observations: A, B, F are collinear; D, C, F are collinear; B, C, E are collinear; A, D, E are collinear.

• Because A, D, E are collinear and B, C, E are collinear, ∠DEC equals the angle between AD and BC (i.e. between the lines AD and BC).

• Because A, B, F are collinear and D, C, F are collinear, ∠BFC equals the angle between BA and CD.

• Therefore the sum is ∠DEC + ∠BFC = (angle between AD and BC) + (angle between BA and CD).

• Rearrange the directed angles: combining the two terms and cancelling the intermediate directions gives ∠BCD − ∠BAD.

Conclusion: ∠DEC + ∠BFC = ∠BCD − ∠BAD.