A parallelogram ABCD is given. The bisector of an acute angle A intersects the side BC at point E.

| April 11, 2021 | education

A parallelogram ABCD is given. The bisector of an acute angle A intersects the side BC at point E. The bisector of an obtuse angle D intersects the side BC at point F. What is the length of the segment EF if AB = 9cm, BC = 10cm?

Since ABCD is a parallelogram, it means AB = CD = 9 cm, BC = AD = 10 cm.
Since AE is a bisector, it means ∠BAE = ∠DAE, and ∠DAE = ∠AEB (as parallel BC and secant AE lying crosswise with AD).
Since ∠ВАЕ = ∠АЕВ, it means that the triangle ABE is isosceles, so AB = BE = 9 cm.
Similarly, ∠CDF = ∠ADF = ∠DFC, triangle СDF is isosceles, then DC = СF = 9cm.
BC = 10 cm, BE = 9 cm, then CE = 10 – 9 = 1 (cm)
ВС = 10 cm, СF = 9 cm, then ВF = 10 – 9 = 1 (cm)
ВС = ВF + FE + EC:

10 = 1 + FE + 1;

FE = 10 – 2 = 8 (cm).

Answer: FE = 8 cm.



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