We study a generalization of the affine preferential attachment model where triangles are randomly added to the graph. We show that the model exhibits an asymptotically power-law degree distribution with adjustable parameter $γ\in (1,\infty)$, and positive clustering. However, the clustering behaviour depends on how it is measured. With high probability, the average local clustering coefficient remains positive, independently of $γ$, whereas the expectation of the global clustering coefficient does not vanish only when $γ>3$.