In Greenberg's (2008) textbook, the following proof that the summit angles of a Saccheri quadrilateral are congruent is given on p. 178.
By hypothesis and SAS, $\triangle DBA \cong \triangle CAB$. Then by SSS, $\triangle DCB \cong \triangle CDA$. Hence $\angle C \cong \angle D$ by angle addition.
The proof is accompanied by the following diagram:
I do see that Greenberg's proof is valid-- although I am surprised that he did not include an argument right before his last sentence showing that the ray $\overrightarrow{CB}$ is between $\overrightarrow{CA}$ and $\overrightarrow{CD}$ and that the ray $\overrightarrow{DA}$ is between $\overrightarrow{DC}$ and $\overrightarrow{DB}$. This argument might take a couple sentences and make the proof longer.
To my mind, a more direct proof would be:
By hypothesis and SAS, $\triangle DBA \cong \triangle CAB$. Then by SSS, $\triangle DCB \cong \triangle CDA$. Hence $\angle C \cong \angle D$ as corresponding angles in the congruent triangles $\triangle DCB$ and $\triangle CDA$.
Is my altered proof valid? If not, what is the error in my reasoning?
(I am just puzzled as to why Greenberg would not take this more direct route-- assuming it is correct.)
Reference
Greenberg, M. J. (2008). Euclidean and Non-Euclidean Geometries: Development and History (4th ed.). New York, NY: W. H. Freedman and Company