Congruence Through Sequences of Rigid Motions

In this lesson:

• Define congruence using sequences of rigid motions
• Describe a sequence that maps one figure onto another
• Decide when two figures are not congruent

Learning Objectives for This Lesson

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

1. Define congruence using a sequence of rigid motions
2. Explain why this beats "same shape and size"
3. Describe a sequence mapping one figure onto another
4. Recognize that transformation order matters
5. Determine when figures are not congruent

Are These Two Triangles the Same?

Different places, pointing different ways — how would you prove they match to someone who can't see your screen?

Recall: The Three Rigid Motions

You already know three moves that slide, flip, and turn figures:

• Translation — slide every point the same way
• Reflection — flip across a line
• Rotation — turn around a center point

Each one keeps every length and every angle exactly the same.

Sliding One Triangle Onto Another

Picture sliding the left triangle five units right.

• Every vertex shifts the same distance, same direction
• It lands exactly on the right triangle
• No stretching needed — just a slide

That single translation is enough here. So the triangles match.

A Precise Definition of Congruence

Two figures are congruent when a sequence of rigid motions maps one exactly onto the other.

• Written
• "Maps exactly onto" means every point lands on its partner
• The test: can you find such a sequence?

Why the Definition Guarantees Equal Measures

Rigid motions never change lengths or angles. So if a sequence maps one figure onto another:

• Every side length must already match
• Every angle measure must already match

"Same shape and size" comes out automatically — for free.

The Symbol and Vertex Correspondence

The order of letters tells you which parts match.

• means , ,
• Side corresponds to side
• Angle corresponds to angle

Order is not optional — it carries information.

A Single Reflection Can Map It

Triangle and its image flip across the -axis.

• , ,
• One reflection lands every vertex — so they're congruent

What If One Move Is Not Enough?

So far, a single slide or flip did the job.

But some congruent figures are slid and turned — or slid and flipped.

No single motion will land one on the other. What now?

When No Single Motion Works

• These triangles are congruent, but they differ in both position and orientation
• A slide alone leaves the orientation wrong; a turn alone leaves it in the wrong place

A Two-Step Strategy That Always Helps

When one motion isn't enough, work in two steps:

1. Translate to land one vertex on its partner
2. Rotate or reflect to swing the other vertices into place

Fix one corner first, then pivot the rest around it.

Step One: Translate to Align a Vertex

• Goal: map onto , aligning with
• Translate 2 left and 2 down:
• Now , — not yet on and

Step Two: Rotate to Finish the Job

• With pinned on , rotate about :
• And — every vertex now matches

Check Every Vertex, and More Than One Works

Always verify all three vertices land correctly.

• ✓, ✓, ✓
• A different translation-then-reflection could also work

There's no single "right" sequence — any valid one counts.

The Order of Moves Matters

A sequence is an ordered list, not a set.

• Translate-then-rotate gives one result
• Rotate-then-translate gives a different result
• Like socks-then-shoes versus shoes-then-socks

Same Moves, Reversed, Different Landing

Take point and apply two moves two ways.

• Up 3, then rotate CCW:
• Rotate first, then up 3:

A Sequence That Needs a Reflection

Not every second step is a rotation — sometimes you must flip.

• and are mirror images, then shifted
• Step 1: translate so lands on
• Step 2: reflect to match the flipped orientation

A rotation alone can never un-mirror a figure.

Your Turn: Find the Sequence

Map onto , sharing vertex .

• and
• Vertex already matches — so no translation needed
• What single rotation about sends and home?

When Can No Sequence Ever Work?

We've been building sequences to show congruence.

But some pairs have no sequence at all.

How could you ever be sure no sequence works — without trying forever?

Different Side Lengths Mean Not Congruent

These triangles look similar, but measure the base.

• One base is , the other is
• Rigid motions never change a length — so can't become

Same Sides but Different Angles

Equal side lengths still aren't enough on their own.

• A rectangle and a slanted parallelogram can share all side lengths
• The rectangle has angles; the parallelogram has and
• Rigid motions preserve angles — so different angles rule it out

Predict: Equal Perimeter Means Congruent?

A -by- rectangle and a -by- rectangle each have perimeter .

Same perimeter — does that make them congruent? Predict before advancing.

Sort Them: Congruent or Not?

For each pair, decide and justify:

1. Two triangles related by a slide
2. One triangle enlarged
3. A pentagon flipped then shifted
4. A triangle and its mirror image

Congruent? Name the moves. Not congruent? Name the difference.

Build the Full Sequence Yourself

Two congruent triangles, no hints, no pre-aligned vertices.

• Pick a vertex to translate first
• Then choose a rotation or reflection to finish
• Write each step in order, with its parameters
• Verify all three vertices land — then state your sequence

How to Prove It, How to Disprove It

✓ Congruent: a sequence of rigid motions maps one figure exactly onto the other

✓ To prove it: describe the sequence, each move in order

✓ To disprove it: find one measurement that differs

"Looks the same" isn't proof; order matters; equal perimeter is not congruence

Coming Up: From Same Size to Similar

Every move today kept the figure the same size.

Next, we allow a new move: dilation — shrinking or stretching a figure.

That one new move is the difference between congruence and similarity.