《2025年诚成教育学业评价七年级数学下册人教版》

24. (8分)如图所示，已知BC//OA，若点E，F在BC上，且∠FOC = ∠AOC，OE平分∠BOF.
(1)试求∠OCB与∠OFB的比；
(2)当∠B = ∠A = 118°，∠OEB = ∠OCA时，试求∠OCA的度数.

答案： 解：
(1)设$\angle FOC = \angle AOC = x°$.
∴$\angle AOF = 2x°$.
∵$BC // OA$，
∴$\angle OFB = \angle AOF = 2x°$，
$\angle OCB = \angle AOC = x°$.
∴$\angle OCB : \angle OFB = 1 : 2$.
(2)设$\angle FOC = \angle AOC = y°$.
∵$\angle B = \angle A = 118°$，$\angle OEB = \angle OCA$，
∴$\angle BOE = \angle AOC = y°$.
∵$OE$平分$\angle BOF$，
∴$\angle EOF = \angle BOE = y°$.
∴$\angle AOB = 4y°$.
∵$BC // OA$，
∴$\angle AOB = 180° - 118° = 62°$.
∴$y = 15.5$.
∴$\angle AOC = 15.5°$.
∵$\angle A = 118°$，
∴$\angle OCA = 46.5°$.