Using Rigid Motions to Define Congruence

HSG.CO.B.6 — Defining and Determining Congruence

In this lesson:

• Define congruence using rigid motions
• Predict and verify the effects of transformations
• Test whether two figures are congruent

Learning Objectives for This Lesson

By the end of this lesson, you should be able to:

1. State the formal definition of congruence in terms of rigid motions
2. Predict the effect of a rigid motion on a figure, including vertex positions and orientation
3. Apply a sequence of rigid motions and verify the image coincides with a target
4. Determine whether two figures are congruent by finding a rigid motion or identifying a measurement mismatch
5. Explain why the rigid-motion definition captures "same shape and size"

What Does "Congruent" Actually Mean?

You've used this word since middle school:

• Triangle ABC and Triangle DEF are congruent
• "Same shape and same size"

But how would you convince a skeptic?

"They look the same" is not a mathematical argument

The Formal Definition of Congruence

• "If and only if" = the implication works both directions
• "Sequence" = one or more translations, reflections, and/or rotations
• "Maps onto" = every point of lands on a corresponding point of

Rigid Motion Mapping: See It Visually

— the rigid motion carries , ,

Why Rigid Motions Guarantee "Same Shape and Size"

Rigid motions preserve all distances and angles:

• Translation: every point moves by the same vector
• Reflection: perpendicular bisector properties keep distances equal
• Rotation: all points stay equidistant from the center

Therefore: congruent figures must have identical side lengths and angle measures

The Two Directions of the Equivalence

Direction 1 (easy): Rigid motion ⟹ same measurements

• Distances and angles preserved → identical measurements guaranteed

Direction 2 (harder): Same measurements ⟹ rigid motion exists

• Translate to align one vertex, rotate to align one side, reflect if needed
• Full proof in CO.B.7

Check-In: Which Statement Uses the Definition Correctly

Which is a valid congruence argument?

A. "△ABC has the same area as △DEF, so they're congruent."

B. "A 90° rotation about the origin maps every vertex of △PQR to the corresponding vertex of △STU."

C. "△ABC looks the same as △DEF in the picture."

Only one uses the formal definition correctly.

Answer: Statement B Is Correct

B uses the formal definition correctly.

• Names a specific rigid motion: 90° rotation about the origin
• Verifies it works: every vertex maps to its target

From now on: proving congruence = finding a rigid motion

Area alone ≠ congruent · Visual similarity ≠ congruent

From the Definition to Geometric Prediction

You now know what congruence means.

Next: can you predict where a rigid motion sends a figure?

• Before computing: predict the quadrant, orientation, and handedness
• Then verify with coordinates

Prediction builds geometric reasoning — not just mechanical calculation

Mental Toolkit: How Translations Affect Figures

What translations do:

• Shift position by a fixed vector
• Every vertex moves the same direction and distance
• Orientation preserved — figure doesn't rotate or flip
• Handedness preserved — clockwise stays clockwise

Quick rule: Add to every vertex coordinate

Mental Toolkit: Reflections and Rotations

Reflections:

• Flip over a line — reverse handedness (orientation changes)
• A clockwise-labeled triangle becomes counterclockwise
• The reflection line is the perpendicular bisector of each vertex-to-image segment

Rotations:

• Rotate about a center point — preserve handedness
• A clockwise-labeled triangle stays clockwise
• Exception: 180° rotation reverses position but preserves orientation

Predict the Image After Reflecting Over the x-Axis

has , , . Reflect over the -axis.

Predict first: Which quadrant? Does orientation flip?

Predict the Image After Rotating 90° Counterclockwise

has , , . Rotate 90° CCW about the origin.

Predict first: Which quadrant? Does handedness change?

Check-In: Predict the Result of Two Steps

has vertices in quadrant II. Apply: (1) reflect over the -axis, (2) translate by .

Predict:

• After step 1: which quadrant?
• After step 2: which quadrant?
• Does the final image have reversed handedness vs. original?

Apply the sequence mentally — then verify on the next slide.

Answer: Two-Step Sequence and Handedness Result

After step 1 (reflect over -axis): image in quadrant I — x-coordinates negated, y unchanged

After step 2 (translate by ): image shifts down 4 units

Handedness vs. original: reversed — the reflection in step 1 flipped orientation

Key insight: Order matters. Reversing the steps gives a different result.

From Predicting to Testing Congruence

You can now predict effects of rigid motions.

Next: use that ability to test congruence

Systematic process:

1. Preliminary check — compare measurements
2. Find a rigid motion (or sequence) mapping one figure to the other
3. Verify every vertex maps correctly

Step 1: Preliminary Measurement Check

: , , · : , ,

Compute side lengths:

Step 1 Completed: Measurements Match

All three sides match: 3, 4, 5. Now find the rigid motion.

Strategy: Translate to , then rotate about to align with

• Direction from to : along positive -axis (angle )
• Direction from to : downward, angle
• Needed rotation: (clockwise) about

Step 2: Find the Rigid Motion

Rigid motion: Translate by , then rotate about

Vector from to : → rotated :

Step 3: Verify Every Vertex

Vector from to : → rotated :

Conclusion: Translate by then rotate about maps

Non-Congruent Figures: When No Rigid Motion Works

Two quadrilaterals — both with sides 3, 4, 3, 4 — but different angles:

• Rectangle: all angles 90°
• Parallelogram: angles 80°, 100°, 80°, 100°

No rigid motion maps one to the other — rigid motions preserve angles.

M2: Visual similarity ≠ congruent
M3: Dilations change distances — not rigid motions

Why Dilation Does Not Establish Congruence

• Dilation (scale factor ): multiplies all distances by — not a rigid motion
• Rigid motion: distances preserved — scale factor always exactly
• Similar figures: same shape, different size → related by dilation, not congruent

Check-In: Find the Rigid Motion

: , , · : , ,

Step 1: Are the side lengths equal? (Check before advancing)

Step 2: If yes, identify a rigid motion mapping to

Step 3: Verify at least two vertices

Write your answer before the next slide

Answer: Reflection over the -axis

Side lengths: , , ✓

Rigid motion: Reflect over the -axis:

Recheck correspondence: , , under reflection

Two Key Clarifications About Congruence Testing

M4: Must you check every point?

No — vertex checking suffices for polygons. Rigid motions preserve collinearity: if and , every point on maps to automatically.

M5: Is the rigid motion unique?

No — many rigid motions may work. The definition requires only one to exist.

From Testing to Properties of Congruence

A relation is an equivalence relation if it satisfies:

• Reflexive: every object relates to itself
• Symmetric: if relates to , then relates to
• Transitive: if relates to and relates to , then relates to

Does congruence satisfy all three? We'll verify directly from the definition.

Reflexive: Every Figure Is Congruent to Itself

Claim: for every figure

Proof: The identity transformation maps every point to itself and preserves all distances and angles — it is a valid rigid motion.

Therefore every figure maps to itself ✓

A translation by vector — moves nothing, still satisfies the definition

Symmetric: Congruence Works in Both Directions

Claim: If , then

Proof: If maps , the inverse maps .

Every rigid motion has an inverse that is also a rigid motion:

• Rotation by → inverse: rotation by
• Reflection → inverse: same reflection (self-inverse)
• Translation by → inverse: translation by

Transitive: Congruence Chains Across Three Figures

Claim: If and , then

Proof: Let map and map . Then the composition maps .

The composition of two rigid motions is a rigid motion:

• Distances preserved by , then again by → distances preserved by ✓

Transitivity in Action: Three-Figure Chain

via reflection over -axis · via translation by

via: reflect over -axis, then translate by

Check-In: Describe the Composed Rigid Motion

Figure has vertices at , , .

Figure via: reflect over the line .

Figure via: translate by .

Describe a rigid motion mapping Figure to Figure .

Write the composition before the next slide.

Answer: Composing the Two Given Rigid Motions

Composition: (1) Reflect over , then (2) translate by

Verify vertex :

— the composition is the rigid motion, no search needed

Key Takeaways and Misconception Warnings

✓ Congruence = rigid motion mapping one figure exactly onto the other
✓ Rigid motions preserve distances and angles → guarantee same shape and size
✓ Test: measure → find rigid motion → verify vertices

Congruent figures may be in different positions and orientations
Visual similarity is not sufficient — find a rigid motion
Dilations are not rigid motions — similar ≠ congruent
One rigid motion is enough — uniqueness not required

What Comes Next: CO.B.7 and Triangle Congruence

HSG.CO.B.7: Equal sides and angles → congruent triangles — proved via rigid motions

Rigid motion proof strategy:

1. Translate to align one pair of vertices
2. Rotate to align one pair of sides
3. Reflect (if needed) to match orientation

HSG.CO.B.8: SAS, ASA, SSS follow as consequences of CO.B.7