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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
monstergreenpoint
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).

Using Rigid Motion(s) to Determine Congruence:
rmnopic1
Is ΔRST congruent 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 congruentΔMNO.

congsingle1
Is ΔABC congruent 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 congruent ΔDEF.
contranscong2
Is parallelogramsymbolPQRS congruent parallelogramsymbolTUVW?

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
parallelogramsymbolPQRS congruent parallelogramsymbolTUVW.
congsingle3
Is ΔEFG congruent 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 congruent ΔJKL.

rmnopic2

Is ΔBCD congruent ΔEFG?
Sometimes a combination of rigid motions is needed to map one figure onto another.

Rigid motions: Reflection and Translation

Assuming bd andef are horizontal, reflect ΔBDC over a horizontal line halfway between bdandef. Then translate the image horizontally to the right to coincide with ΔEFG making ΔBCD congruentΔEFG.
congsingle4
Is ΔABC congruent 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 congruentΔDEF.


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