Which transformation would result in identical shapes with the same measures?

The correct choice is isometries because isometric transformations preserve the size and shape of figures. When a shape undergoes an isometric transformation, such as a rotation, reflection, or translation, the image maintains the same dimensions and angles as the original figure. This means that not only are the corresponding lengths of the sides of the shapes equal, but the orientation and shape also remain unchanged, making the two figures congruent.

In contrast, dilation and contraction involve changes in size—while they may maintain the proportions of the shapes, they do alter the measurements, resulting in similar but not identical figures. Translation, while it does move a shape, does not change its size or orientation, but isometries specifically refer to transformations that keep all characteristics of the shape intact throughout.