# The bisectors of the angles A and D of the parallelogram ABCD intersect at a point lying on the side BC. Find AB if BC = 34

May 31, 2020 | Education

| BC || AD (by definition of parallelogram)

∠BAE = ∠EAD (since AE is a bisector)

∠EAD = ∠BEA (as these are cross-angles)

Therefore, ∠BAE = ∠BEA

It turns out that the triangle ABE is an isosceles (by property), and AB = BE (by the definition of an isosceles triangle).

Similarly with the triangle ECD:

∠CED = ∠CDE

EC = CD

Since AB = CD (by the parallelogram property), it turns out that AB = BE = EC = CD.

Therefore, BE = BC / 2 = 34/2 = 17.

AB = BE = 17

Answer: AB = 17

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