Two figures are congruent if and only if there exists one, or more, rigid motions which will map one figure onto the other.

Rigid motions are:   • reflections   • translations   • rotations

Rigid motions are also called:   • rigid transformations   • isometries   • congruence transformations

Rigid motions move figures to a new location without altering their size or shape
(thus maintaining the conditions for the figures to be congruent).

	Is ΔRST ΔMNO? By the definition of congruent, we need to find a rigid motion that will map ΔRST onto ΔMNO. Rigid motion: Reflection A reflection over line l will map ΔRST to coincide with ΔMNO, making ΔRST ΔMNO.
	Is ΔABC ΔDEF? By the definition of congruent, we need to find a rigid motion that will map ΔABC onto ΔDEF. Rigid motion: Reflection A reflection over the y-axis will map ΔABC to coincide with ΔDEF, making ΔABC ΔDEF.
	Is PQRS TUVW? We need to find a rigid motion that will map one parallelogram onto the other. Rigid motion: Translation The translation (x, y) → (x + 4, y - 4) will map PQRS onto TUVW, making PQRS TUVW.
	Is ΔEFG ΔJKL? We need to find a rigid motion that will map one triangle onto the other. Rigid motion: Rotation A rotation of 90º about the origin will map ΔEFG to coincide with ΔJKL, making ΔEFG ΔJKL.
	Is ΔBCD ΔEFG? Sometimes a combination of rigid motions is needed to map one figure onto another. Rigid motions: Reflection and Translation Assuming and are horizontal, reflect ΔBDC over a horizontal line halfway between and. Then translate the image horizontally to the right to coincide with ΔEFG making ΔBCD ΔEFG.
	Is ΔABC ΔDEF? We need to find a combination of rigid motions that will map one triangle onto the other. Rigid motions: Reflection and Translation A reflection in the x-axis, followed by a translation of (x, y) → (x - 6, y + 1), will map ΔABC to coincide with ΔDEF, making ΔABC ΔDEF.