In a triangle ABC-bisector BD, angle A = 75 degrees, angle C = 35 degrees. Prove that triangle BDC is isosceles.

By the condition of the problem, we know the degree measures of the two angles of the triangle ABC. Find the unknown angle B:
∠ В = 180 ° – (∠А + ∠С) = 180 ° – 110 ° = 70 °.
BD by the condition the bisector of angle B, we obtain that the angle DBC is equal to:
∠ DBC = 1/2 * ∠ B = 35 °.
In the DBC triangle, we have two angles with the same degree measure, each 35 °. We conclude that the DBC triangle is isosceles. Q.E.D.
Answer: triangle DBC is isosceles.