There's only one question; you are confused.
[tex] \angle CXY \cong \angle BXY \qquad \textrm{Given}[/tex]
[tex] \angle CXA \cong \angle BXA \qquad \textrm{Supplements of congruent angles are congruent}[/tex]
[tex] \overline{AX} \cong \overline{AX} \qquad \textrm{Reflexivity of congruency}[/tex]
[tex] \angle CAX \cong \angle BAX \qquad \textrm{Given}[/tex]
[tex] \triangle CAX \cong \triangle BAX \qquad \textrm{ASA}[/tex]
[tex] \overline{CX} \cong \overline{BX} \qquad \textrm{Corresponding parts}[/tex]
[tex] \overline{XY} \cong \overline{XY} \qquad \textrm{Reflexivity of congruency}[/tex]
[tex]\triangle XCY \cong \triangle XBY \qquad \textrm{SAS}[/tex]
[tex] \angle XCY \cong \angle XBY \qquad \textrm{Corresponding parts}[/tex]
