What Each Transformation Preserves

Translations, reflections, and rotations are called rigid motions because they move a figure without stretching or squishing it. Pick any two points on the figure and measure the distance between them before and after the move — it's unchanged. Same for any angle. That's why the output is always congruent to the input: same size, same shape, just relocated or re-oriented. A dilation breaks that rule on purpose. It keeps every angle the same (so the shape looks identical) but multiplies every length by a scale factor, shrinking or stretching the figure into a similar copy rather than a congruent one.

Orientation is the one property that splits the three rigid motions into two camps. A translation or rotation keeps the vertex order the same — read a triangle's vertices clockwise before the move, and they're still clockwise after. A reflection flips that order every single time, clockwise becomes counterclockwise, because you're looking at the mirror image. This reversal has zero effect on area or angle measure, since area just counts how much space is enclosed and that space doesn't change when you flip the figure over.

This is the idea that separates congruence transformations from similarity transformations elsewhere in the unit. If a problem gives you two figures with identical angles but different side lengths in the same ratio, you're looking at a dilation. If every length and angle matches exactly, you're looking at some combination of slides, flips, and turns, and the only question left is which one, which often comes down to checking whether orientation reversed.