For any chain map $f : A^\bullet \to B^\bullet$, the triangle $A \xrightarrow{f} B \xrightarrow{\iota} \mathrm{Cone}(f) \xrightarrow{p} A[1]$ is distinguished in $K(\mathcal{A})$. Here $\mathrm{Cone}(f)^n = A^{n+1} \oplus B^n$ with differential $d(a, b) = (-da, f(a) + db)$.

In $D(\mathcal{A})$, a short exact sequence $0 \to A \to B \to C \to 0$ in $\mathcal{A}$ gives a distinguished triangle $A \to B \to C \xrightarrow{\delta} A[1]$. The morphism $\delta$ corresponds to the extension class in $\mathrm{Ext}^1(C, A)$.

The triangle $A \to A \oplus B \to B \to A[1]$ with the obvious maps and $w = 0$ is distinguished. This corresponds to the split short exact sequence. The zero connecting homomorphism means the extension is trivial.

In the stable homotopy category, for $f : X \to Y$, the cofiber sequence $X \xrightarrow{f} Y \to Y/X \to \Sigma X$ is a distinguished triangle. The shift $[1]$ is the suspension $\Sigma$.

If $X \xrightarrow{u} Y \xrightarrow{v} Z \xrightarrow{w} X[1]$ is distinguished, then $Y \xrightarrow{v} Z \xrightarrow{w} X[1] \xrightarrow{-u[1]} Y[1]$ is also distinguished. The sign $-u[1]$ is essential.

Applying $\mathrm{Hom}(W, -)$ to a distinguished triangle $X \to Y \to Z \to X[1]$ gives the long exact sequence:

$\cdots \to \mathrm{Hom}(W, X[n]) \to \mathrm{Hom}(W, Y[n]) \to \mathrm{Hom}(W, Z[n]) \to \mathrm{Hom}(W, X[n+1]) \to \cdots$

In $D(\mathcal{A})$, a complex $C^\bullet$ is acyclic iff $C^\bullet \cong 0$ iff $0 \to C^\bullet \to 0 \to 0[1]$ is a distinguished triangle iff $C^\bullet$ sits in a distinguished triangle $C \to 0 \to ? \to C[1]$.

$\mathrm{Ext}^n(X, Y) = \mathrm{Hom}_{D(\mathcal{A})}(X, Y[n])$. A class $\xi \in \mathrm{Ext}^1(C, A)$ corresponds to a distinguished triangle $A \to B \to C \xrightarrow{\xi} A[1]$ where $B$ is the extension.

For an element $x \in R$ and an $R$-module $M$, the Koszul triangle is $M \xrightarrow{x} M \to M/xM \to M[1]$ in $D(R)$. This is the triangle associated to the short exact sequence $0 \to M \xrightarrow{x} M \to M/xM \to 0$ (when $x$ is a non-zero-divisor on $M$).

If $X_i \to Y_i \to Z_i \to X_i[1]$ are distinguished for $i = 1, 2$, then $X_1 \oplus X_2 \to Y_1 \oplus Y_2 \to Z_1 \oplus Z_2 \to (X_1 \oplus X_2)[1]$ is distinguished. Distinguished triangles are preserved by direct sums.

In the Verdier quotient $\mathcal{T}/\mathcal{S}$, a triangle is distinguished iff it is isomorphic to the image of a distinguished triangle in $\mathcal{T}$. The localization functor $\mathcal{T} \to \mathcal{T}/\mathcal{S}$ is exact (preserves distinguished triangles).

For a distinguished triangle $X \to Y \to Z \to X[1]$ in $D^b(\mathcal{A})$, $\chi(Y) = \chi(X) + \chi(Z)$, where $\chi = \sum (-1)^n [H^n]$ in the Grothendieck group.