I am reading the section on Strainers in Burago, Burago and Ivanov's book "A Course in Metric Geometry". I have been struggling with the proofs of some of the lemmas.
On Lemma 10. 8. 13, the authors say that having $|pq|<\frac{\varepsilon}{4}|pa|$ implies that $\widetilde{\angle}paq< \arcsin \frac{\varepsilon}{4} < \frac{\varepsilon}{2}$. I have been trying to find hoplessly a way to prove this but nothing seems to work for me. I've tried applying both sine and cosine laws in the case $k=0$ where $\mathrm{Curv} (X)\geq k$ on the comparison triangles.
On the next Lemma. 10.8.14, they use the inequality
$|xz|<|xy|+|yz|\sin(\angle(xyz)-\frac{\pi}{2})$
for a triangle in $\mathbb{R}^2$. While I have been able to give a proof of this inequality by following the hint they give, I have not been able to produce analog versions for the other space forms. I feel like this proof is only valid for curvature not less than $0$.
Finally on Proposition 10.8.15, they use the facts that $|a_ix_n>1|$ and $|x_nx_{n+1}|<2\delta$ to conclude that $\widetilde{\angle}x_na_ix_{n+1}<4\delta<\varepsilon$. Can anyone give me some hint to see how is this achieved?
I realize it is inconvenient to ask this without transcribing the proofs here to give more context to the question but the proofs are very long and doing so would take me forever.
your first Question:
I have asked myself the same and got an answer on stackexchange (Linear bound on angles in an euclidean triangle.).
your second Question:
The proof is carried out for spaces of curvature $\geq k$. But since the statements are local, one can achieve this situation via rescaling (the statement before 10.8.2 on page 380 in the book). So it does not matter whether the formula is not true anymore in spaces of constant curvature -k.