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We study the $D_s^+\to \pi^+(a_0(980)^0\to)\pi^0\eta$,$\pi^0(a_0(980)^+\to)\pi^+\eta$ decays, which have been recently measured bythe BESIII collaboration. We propose that $D_s^+\to\pi^{+(0)}(a_0(980)^{0(+)}\to)\pi^{0(+)}\eta$ receives the contributions fromthe triangle rescattering processes, where $M^0$ and $\rho^+$ in $D_s^+\to M^0\rho^+$, by exchanging $\pi^{0(+)}$, are formed as $a_0(980)^{0(+)}$ and$\pi^{+(0)}$, respectively, with $M^0=(\eta,\eta')$. Accordingly, we calculatethat ${\cal B}(D_s^+\to a_0(980)^{0(+)}\pi^{+(0)})=(1.7\pm 0.2\pm 0.1)\times10^{-2}$ and ${\cal B}(D_s^+\to\pi^{+(0)}(a_0(980)^{0(+)}\to)\pi^{0(+)}\eta)=(1.4\pm 0.1\pm 0.1)\times10^{-2}$, being consistent with the data.

Resonant $a_0(980)$ state in triangle rescattering $D_s^+\to \pi^+\pi^0\eta$ decays