Transformational Definition:
def
Two figures are congruent if and only if there exists one, or more, rigid transformations which will map one figure onto the other.

Rigid transformations are:
 
• reflections
 
• translations
 
• rotations
monstergreenpoint
Rigid transformations can be called:
 
• isometries
 
• congruence transformations
 
• rigid motions

Rigid transformations move figures to a new location without altering their size or shape
(thus maintaining the conditions for the figures to be congruent).
If figures are congruent, the corresponding sides are congruent,
and the corresponding angles are congruent.

To show that figures are congruent using rigid transformations,
you must establish a way to map the pre-image figure onto the resulting image
using rigid transformations.

Using Rigid Transformation(s) to Determine Congruence:
rmnopic1
Given the triangles positioned and marked as shown with a set of sides being parallel.
Is ΔRST congruent ΔMNO?
Consider ΔRST to be the pre-image.

By the definition of congruent, we need to find a rigid transformation that will map ΔRST onto ΔMNO.

Rigid transformation: Reflection

Establish a reflection line l half-way between the triangles, as shown, over which ΔRST can be mapped to coincide with ΔMNO.
Thus, ΔRST congruentΔMNO.
congsingle1
Given the graph at the left.
Is ΔABC congruent ΔDEF?
Consider ΔABC to be the pre-image.

By the definition of congruent, we need to find a rigid transformation that will map ΔABC onto ΔDEF.

Rigid transformation: Reflection

A reflection over the y-axis will map ΔABC to coincide with ΔDEF, making
ΔABC congruent ΔDEF.
contranscong2
Given the graph at the left.
Is parallelogramsymbolPQRS congruent parallelogramsymbolTUVW?
Consider parallelogram PQRS to be the pre-image.

We need to find a rigid transformation that will map one parallelogram onto the other.

Rigid transformation: Translation

The translation (x, y) → (x + 4, y - 4) will map PQRS onto TUVW, making
parallelogramsymbolPQRS congruent parallelogramsymbolTUVW.
congsingle3
Given the graph at the left.
Is ΔEFG congruent ΔJKL?
Consider ΔEFG to be the pre-image.

We need to find a rigid transformation that will map one triangle onto the other.

Rigid transformation: Rotation

A rotation of 90º CCW about the origin will map ΔEFG to coincide with ΔJKL, making
ΔEFG congruent ΔJKL.

rmnopic2
Given the diagram at the left.
Is ΔBCD congruent ΔEFG?
Consider ΔBCD to be the pre-image.

Rigid transformations: Reflection and Translation (Sometimes a combination of rigid transformations is needed to map one figure onto another.)

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
Given the graph at the left.
Is ΔABC congruent ΔDEF?
 Consider ΔABC to be the pre-image.

We need to find a combination of rigid transformations that will map one triangle onto the other.

Rigid transformations: 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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