Proof similarity of triangles?

Question

my task is to calculate $\sphericalangle CXY, \sphericalangle CYX, \sphericalangle XCY$. My instructor gave me the hint to proof (with basic geometric knowledge) $\triangle ABC \simeq \triangle AHC \simeq BHC$. If that is shown, it follows $\sphericalangle BAC = \alpha = \sphericalangle HXY = \sphericalangle HCB$ and the same for the $\beta$'s. Can someone explain to me, for example why $\sphericalangle HXY = \alpha$. From that point its indeed obvious that $\triangle ABC \simeq \triangle AHC \simeq BHC$?

Answer 1

Referring to the image in the OP, it is easily seen that $\triangle ABC \simeq \triangle AHC \simeq \triangle BHC$ since all three triangles have the same angles: alpha, beta and a right angle.