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We address the dynamics of two-dimensional (2D) truncated Airy waves and three-component solitons in the system of two fundamental-frequency and second-harmonic fields, coupled by quadratic (chi^(2)) terms. The system models second-harmonic-generating optical media and atomic-molecular mixtures in Bose-Einstein condensates. In addition to stable solitons, the system maintains truncated-Airy-waves states in either one of the fundamental-frequency components, represented by exact solutions, which are stable, unlike the Airy waves in the degenerate (two-component) chi^(2) system. It is also possible to imprint vorticity onto the 2D Airy modes. By means of systematic simulations, we examine interactions between truncated Airy waves originally carried by different fundamental-frequency components which are bending in opposite directions, through the second-harmonic field. The interaction leads to fusion of the input into a pair of narrow solitons. This is opposed to what happens in the 1D system, where the interacting Airy waves split into a large number of solitons. The interaction of truncated Airy waves carrying identical imprinted vorticities creates an additional pair of solitons, while opposite vorticities create a set of small-amplitude "debris" in the output. Slowly moving solitons colliding with a heavy truncated Airy wave bounce back, faster ones are absorbed by it, and collisions are quasi-elastic for fast solitons. Soliton-soliton collisions lead to merger into a single mode, or elastic passage, for lower and higher velocities, respectively.