I'm from Italy, so I was reading an Italian translation of Euclid's Elements; and while I was doing just that, something about the introduction—written by Attilio Frajese—really caught my attention.
Frajese explains how Euclid, not feeling confident about his own parallel postulate, tried to delay its usage as much as he could, wanting to see how far he could push geometry forward before he was forced to make use of it. In particular, he focuses on the very existence of proposition $\text{I}, 17$, which states that the sum of any two angles within any triangle is always less than $180^°$ (in modern notation, of course). This proposition is apparently useless and redundant, seeing how it's just a consequence of $\text{I}, 32$—the much more well-known theorem according to which the sum of the internal angles of a triangle is precisely $180^°$.
This apparent redundancy is easily explained by Frajese as he points out that between these two theorems is $\text{I},29$, the first proposition that Euclid cannot prove without his postulate parallel. Therefore, $\text{I}, 17$ exists as a separate proposition simply because Euclid wanted to write down as many results as he could that didn't depend on that postulate, even if they're mere corollaries of results he will later prove in a more general way.
But then, this leads me to a question. If $\text{I}, 17$ is indeed independent from the parallel postulate, how can elliptic geometry be consistent? The independence of the fifth postulate from the other ones is usually shown by exhibiting geometries like this, which make use of the first four postulates while replacing the fifth one with something else, and then pointing out that the resulting system is still consistent.
But in elliptic geometry a triangle will look like this thing here—see image below; and surely the angles in this triangle add up to more than $180^°$ (indeed, it's one of the most peculiar traits of elliptic geometry that everyone knows). So how is this not in violation of the (parallel-postulate-free) proposition $\text{I}, 17$?
Proposition I, $17$ is proved using Proposition I, $16$ (see here). Proposition I, $16$ doesn’t hold in elliptic geometry. This is explained on this page – I won’t reproduce the entire argument here, but in summary, the proof of Proposition I, $16$ asks you to extend $PS$ to $T$ so that $|PS|=|ST|$ (so in your example $T$ would be the pole opposite to $P$), and to extend $PR$ to $U$, and it then claims that the angle $SRU$ is greater than the angle $SRT$, apparently because $RT$ is expected to lie “within” the angle $SRU$, but that’s not the case here.
Thus, Euclid’s proof is in this case not purely based on the postulates (since the first four postulates hold in elliptic geometry and the fifth hasn’t been used).