Lesson 4.2.4

Defining Congruence via Rigid Isometries

Two figures are congruent if and only if there exists a sequence of rigid motions (isometries) that maps one onto the other. This is the modern, transformation-based definition of congruence.

Introduction

In earlier courses, “congruent” meant “same shape and size.” Now we have the precise tool to define it: two figures are congruent if you can move one onto the other using translations, reflections, rotations, and glide reflections — preserving every distance and angle.

Past Knowledge

All four rigid motions (4.1–4.2.3). Compositions (4.2.2).

Today's Goal

Use rigid motions to prove or disprove congruence between figures.

Future Success

Triangle congruence proofs (Unit 5) and similarity via dilations (Unit 7).

Key Concepts

Congruence (Transformation Definition)

Figure is congruent to Figure (written ) if and only if there exists a sequence of rigid motions that maps onto .

What Rigid Motions Preserve

  • Distances — segment lengths stay the same
  • Angle measures — all angles keep their size
  • Parallelism — parallel sides stay parallel
  • Betweenness — if B is between A and C, it stays between them

How to Prove Congruence

  1. Identify corresponding vertices between the two figures.
  2. Find a rigid motion (or sequence) mapping each vertex of the pre-image to the corresponding vertex of the image.
  3. Verify that the transformation works for all points, not just vertices.

Disproving Congruence

If any distance or angle changes, the figures are not congruent. Find one pair of corresponding sides with different lengths — that's enough.

Worked Examples

Basic

Proving Congruence via Translation

Show that with , , is congruent to with , , .

Compare corresponding points: : , shift = .

Check: ✓

✓

The translation maps every vertex of to the corresponding vertex of .

via translation .

Vector
⟨,⟩
Adjust the vector to explore
A(1,1)→A′(4,3)
B(4,1)→B′(7,3)
C(1,5)→C′(4,7)
Intermediate

Proving Congruence via Reflection

Prove with , , is congruent to with , , .

Each -coordinate is negated while stays the same. This is the rule — reflection across the x-axis.

Since reflection is a rigid motion, it preserves distances and angles.

via reflection across the x-axis.

Reflect across
P(2,3)→P′(2,-3)
Q(5,3)→Q′(5,-3)
R(3,6)→R′(3,-6)
Advanced

Disproving Congruence

Is with , , congruent to with , , ?

Compute side lengths:

Since , no rigid motion can map one to the other (rigid motions preserve distance).

— side lengths differ.

A(0,0)→A′(0,0)
B(3,0)→B′(3,0)
C(0,4)→C′(0,4)

Common Pitfalls

Saying “They Look the Same” Without Proof

Congruence must be proven by specifying the rigid motion. “They look congruent” is not a proof — you need to name the transformation and verify it.

Matching Vertices in the Wrong Order

means , , . Getting the correspondence wrong leads to false conclusions.

Confusing Congruent with Similar

Similar figures have the same shape but possibly different sizes (they use dilations). Congruent figures must have the same shape and same size — only rigid motions (no scaling).

Real-Life Applications

Manufacturing & Quality Control

Every manufactured part must be congruent to the blueprint. Inspectors use coordinate measuring machines (CMMs) that check if rigid motions can align a part to its CAD model — the same concept as our transformation-based congruence definition.

Jigsaw Puzzles

Each puzzle piece fits into exactly one spot. You find its home by applying translations and rotations until it aligns — you're using rigid motions to check congruence with the hole's shape.

Practice Quiz

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