# In triangle ABC, heights AA1, BB1 and CC1 are drawn. Find the angles of the triangle ABC

In triangle ABC, heights AA1, BB1 and CC1 are drawn. Find the angles of the triangle ABC if it is known that the angles of the triangle A1B1C1 are 30, 60, 90 degrees.

Consider a triangle ABC, it is arbitrary, draw the heights AA1, BB1 and CC1. AA1⊥CB, BB1⊥AC, CC1⊥AB.
By the property of the heights of the triangle, if two heights AA1 and BB1 are drawn in the triangle, then the triangles A1B1C and ABC = are similar, which means their corresponding sides are similar, and the angles are equal.
BC / B1C1 = k.
Consider the heights AA1 and CC1, then the triangles A1BC1 and ABC = are similar, which means their corresponding sides are similar, and the angles are equal.
AC / A1C1 = k.
Consider the heights BB1 and CC1, then the triangles AB1C1 and ABC = are similar, which means their corresponding sides are similar, and the angles are equal.
AB / A1B1 = k.
AB / A1B1 = k = AC / A1C1 = BC / B1C1.
So triangles ABC and A1B1C1 are similar, and their corresponding angles are equal, then the angles of triangle ABC are equal to 30, 60, 90 degrees.
Answer: the angles of the triangle ABC are equal to 30, 60, 90 degrees.

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