In the figure below, β π·πΈπΆ + β π΅πΉπΆ is equal to ____________ .
2018
In the figure below, β π·πΈπΆ + β π΅πΉπΆ is equal to ____________ .
- A.
β π΅πΆπ· β β π΅π΄π·
- B.
β π΅π΄π· + β π΅πΆπΉ
- C.
β π΅π΄π· + β π΅πΆπ·
- D.
β πΆπ΅π΄ + β π΄π·πΆ
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Correct answer: A
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, β 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, β BFC equals the angle between BA and CD.
Therefore the sum is β DEC + β 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 β BCD β β BAD.
Conclusion: β DEC + β BFC = β BCD β β BAD.
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