a right triangle ABC with ∠B a right angle, a line segment B'C' connects side AB with the hypotenuse so that ∠AB'C' is a right angle as shown. Use facts about similar triangles to show why (C') =(C).

a right triangle ABC with ∠B a right angle, a line segment B'C' connects side AB with the hypotenuse so that ∠AB'C' is a right angle as shown. Use facts about similar triangles to show why (C') =(C).

I assume that by (C')=(C) you mean that angles \(\displaystyle \angle ACB\cong\angle AC'B'\).
Because \(\displaystyle \overline{BC}\bot\overline{AB}~\&~\overline{B'C'}\bot\overline{AB}\) then we have corresponding angles of parallel lines.