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Congruence via
Rigid Motions

Master the connection between translations, reflections, and rotations and the formal definition of congruent figures.

Geometry

What Are Rigid Motions?

A rigid motion (also called an isometry) is a transformation that moves every point of a figure the same way, preserving both distances and angle measures. The figure's shape and size remain unchanged: only its position or orientation may differ.

The Three Rigid Motions

Translation
Slides every point the same distance in the same direction. No turning, no flipping.
Reflection
Flips the figure across a line of reflection, producing a mirror image.
Rotation
Turns the figure about a fixed center point through a specified angle.

Visualizing the Three Transformations

Translation A B C A' B' C' Reflection D E F mirror D' E' F' Rotation G H I center G' H' I'

The Definition of Congruence

This is the modern, transformation-based definition. Instead of checking every side and angle pair individually, you show that a rigid motion (or a composition of several) carries one figure onto the other. If such a mapping exists: congruent. If no such mapping is possible: not congruent.

Core Theorem: Rigid Motions Preserve Congruence

A B C 5 cm 5 cm 6 cm ≅ A' B' C' 5 cm 5 cm 6 cm

What Rigid Motions Preserve

1
Distance: The length of every segment stays the same: \( d(T(P), T(Q)) = d(P, Q) \).
2
Angle measure: Every angle in the image has the same measure as its preimage angle.
3
Parallelism: Parallel lines remain parallel after the transformation.
4
Betweenness: If point \( B \) is between \( A \) and \( C \), then \( T(B) \) is between \( T(A) \) and \( T(C) \).

Congruence vs Similarity

Congruence Similarity
Transformations used Rigid motions only Rigid motions + dilations
Preserves distances? Yes No (scales them)
Preserves angles? Yes Yes
Same shape? Yes Yes
Same size? Yes Not necessarily
Notation \( \cong \) \( \sim \)

How to Prove Congruence

1
Identify the two figures and label their corresponding vertices.
2
Find a rigid motion (or sequence) that maps one figure onto the other. Start with a translation to align one pair of corresponding points.
3
Apply a rotation (if needed) to align a second pair of corresponding points.
4
Apply a reflection (if needed) to match orientation and align the third point.
5
Conclude: since a sequence of rigid motions maps the first figure exactly onto the second, the figures are congruent.

Example 1: Translation

P Q R 6 6 translate right P' Q' R' 6 6

Example 2: Reflection

D E F 5 5 6 line of reflection D' E' F' 5 5 6

Example 3: Composition of Motions

J K L original translate after translation
rotate \(90^\circ\) CCW
 J' K' L' final image

Predicting the Effect of a Rigid Motion

Given a figure and a specific rigid motion, you can predict exactly where every point ends up using coordinate rules:

Translation
\( (x, y) \to (x + a, y + b) \)
Reflection over \( y \)-axis
\( (x, y) \to (-x, y) \)
Rotation \(90^\circ\) CCW about origin
\( (x, y) \to (-y, x) \)

When Figures Are Not Congruent

A B C 3 3 4 ≇ D E F 6 6 8

Common Mistakes

Confusing Congruent with Similar
Congruent means same shape and same size. Similar only requires the same shape. A dilation is not a rigid motion.
Forgetting Reflections Count
A mirror image is congruent to the original. Don't say "not congruent" just because the figure is flipped.
Incorrect Vertex Mapping
When writing \( \triangle ABC \cong \triangle DEF \), the order matters: \( A \to D \), \( B \to E \), \( C \to F \). Scrambled letters give wrong correspondences.

Question 1 of 5

Which of the following transformations is not a rigid motion?

Question 2 of 5

Triangle \( \triangle ABC \) has vertices \( A(1, 2) \), \( B(4, 2) \), \( C(4, 6) \). After applying the transformation \( (x, y) \to (x + 3, y - 1) \), is the image congruent to the original?

Question 3 of 5

Two figures are congruent. Which statement must be true?

Question 4 of 5

Point \( P(2, 5) \) is rotated \(90^\circ\) counterclockwise about the origin. What are the coordinates of \( P' \)?

Question 5 of 5

\( \triangle ABC \) has sides of length 5, 7, and 9. \( \triangle DEF \) has sides of length 10, 14, and 18. Are the triangles congruent?