Let's break down each task in the problems provided:

Problem 1:

Given:

• $KA$ is perpendicular to the plane of triangle $ABC$.
• $KB \perp BC$.

Tasks:

• (a) Prove that triangle $ABC$ is a right triangle.
• (b) Prove that $KA$ is perpendicular to the planes $KAC$ and $ABC$.
• (c) Find $KA$, given that:
  • $AC = 13$ cm
  • $BC = 5$ cm
  • $\angle KBA = 45^\circ$

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Problem 2:

Given:

• The base $AC$ of an isosceles triangle lies on a plane $\alpha$.
• The angle between the planes $ABC$ and $\alpha$ is given.

Tasks:

• Find the distance from point $B$ to the plane $\alpha$, given that:
  • $AB = 20$ cm
  • $AC = 24$ cm
  • The dihedral angle between the planes $ABC$ and $\alpha$ is $30^\circ$.

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Problem 3:

Given:

• Lines $AB$ and $AC$ are inclined from point $A$ to the plane $\alpha$, forming equal angles with the plane.
• It is known that $BC = AB$.

Task:

• Find the angles of triangle $ABC$.

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I will now begin solving these problems one by one, starting with the first. Let me know if you would like any specific explanation or step-by-step detail for any task!