Triangle Congruence via Rigid Motions

HSG.CO.B.7 — Proving Triangles Congruent

In this lesson:

• Prove the biconditional connecting rigid motions to matching parts
• Derive CPCTC from rigid-motion preservation
• Construct the rigid motion from six matching parts

This lesson proves the theorem that bridges the rigid-motion and measurement-based definitions of congruence.

Learning Objectives for Today's Lesson

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

1. State the biconditional: iff all 6 corresponding pairs match
2. Prove the forward direction: rigid motion implies all corresponding parts are congruent
3. Prove the reverse direction: construct the rigid motion from six matching parts
4. Apply CPCTC to deduce specific congruences from established triangle congruence
5. Build an explicit rigid-motion sequence mapping one triangle onto a congruent triangle

From CO.B.6 to CO.B.7: What We Still Need

You already know from CO.B.6:

• Congruence = a rigid motion maps one figure onto the other
• Rigid motions preserve distances and angles

The open question: Does "congruent via rigid motions" mean exactly the same thing as "all six parts match"?

The Biconditional: Two Separate Claims

• Forward: Rigid motion exists ⟹ all 6 pairs match
• Reverse: All 6 pairs match ⟹ rigid motion exists
• Both directions require independent proofs

Correspondence Diagram Showing Color-Coded Vertex Pairs

The notation encodes the correspondence: first vertex to first, second to second, third to third.

Proof Strategy: Forward vs. Reverse Directions

Forward direction (Chunk 2 — straightforward):

• Uses what you already know: rigid motions preserve distances and angles

Reverse direction (Chunk 3 — the construction):

• Requires building a rigid motion from scratch
• Three steps: translate, rotate, reflect if needed
• Relies on a special property of triangles

Check-In: Read the Congruence Notation

Which vertex maps to which? Write the three vertex pairs.

Also: Which side is claimed congruent to ? To ?

The notation encodes the correspondence — read letter by letter.

Answer: Reading Correspondence from Congruence Notation

tells us:

• , ,
• , ,
• , ,

The third letter corresponds to the third letter — order is everything.

M3: Correspondence comes from the rigid motion — not the alphabet.

Forward Direction: Rigid Motion Implies Matching Parts

If rigid motion maps onto (with , , ):

These follow from CO.A.4: rigid motions preserve all distances and all angle measures.

Side Congruence Follows from Distance Preservation

By CO.A.4: rigid motions preserve all distances between point pairs.

• , ⟹
• , ⟹

Angle Congruence and the CPCTC Result

By CO.A.4: rigid motions preserve all angle measures.

CPCTC — Corresponding Parts of Congruent Triangles are Congruent:

Not a magic rule — a direct consequence of rigid-motion preservation.

CPCTC: Six Congruences at Once

• From one rigid motion, you get six congruences for free
• Three side pairs + three angle pairs
• Use whichever ones your proof needs

Check-In: Applying the Forward Direction

A rigid motion maps onto , with , , .

1. Name all three pairs of congruent sides.
2. Name all three pairs of congruent angles.
3. Is ? Explain why or why not.

Pause: work through all three questions before the next slide.

Answer: All Six Pairs from CPCTC

From (with , , ):

• Sides: , ,
• Angles: , ,
• Is ? Not guaranteed — and

M5: CPCTC gives all 6 pairs — sides AND angles, both.

Reverse Direction: What We Need to Prove

Claim: If all six pairs match, a rigid motion exists mapping onto .

Why is this harder than the forward direction?

• Forward: rigid motion is given — just apply preservation
• Reverse: must construct the rigid motion from scratch

Strategy: Three steps — translate, rotate, reflect if needed.

Reverse Direction Step 1: Translate

Goal: Map vertex onto vertex .

Apply the translation that sends to .

• Call the image: where
• Translation preserves all side lengths and angles
• , so is at distance from

Reverse Direction Step 2: Rotate

Goal: Map onto while keeping fixed.

• ✓
• , so is at distance from
• Rotate about by the angle from ray to ray

This maps exactly onto . Now: and ✓

Step 2 Result: Two Vertex Pairs Aligned

After Step 2: , — two vertex pairs aligned.

Where is ?

Check-In: Locating C-Double-Prime

After Steps 1–2: , .

Where must be located?

• → lies on a circle of radius centered at
• → lies on a circle of radius centered at

Question: How many points satisfy both conditions? Where are they?

Think geometrically before the next slide.

Reverse Direction Step 3: Reflect if Needed

Two circles centered at and intersect in at most two points — one on each side of line .

• is at one intersection point; is at the other (or the same)
• If : done — no reflection needed
• If : reflect over line

Reflecting over maps to while keeping and fixed.

Step 3 Result: Construction Complete

At most 3 rigid motions (translate, rotate, reflect) map onto . ✓

Coordinate Example: Verifying All Six Parts Match

: , , · : , ,

Execute Step One: Translate Vertex A onto Vertex D

Translation vector: — moves every vertex by .

is at distance 5 from — the same distance as .

Execute Step Two: Rotate to Place B-Prime on E

, , .

Vector . Rotating 90° CCW gives .

maps to under the same rotation — not .

Execute Step Three: Reflect over Line DE

After Steps 1–2: and .

Line is the vertical line .

The construction maps onto in three rigid motions.

Triangle Rigidity: Three Vertices Determine the Shape

A triangle is rigid — three vertices determine the entire figure.

• Once , , are fixed, all sides and angles are fully determined
• No "flexing" possible — the shape cannot change with fixed vertices

Why this matters: After matching three vertices (, , ), the entire triangle is matched.

Check-In: Does Triangle Rigidity Extend to All Polygons?

You can build a physical triangle from three rigid sticks — it cannot flex.

Question: Can you build a quadrilateral from four rigid sticks that flexes?

Also: Why does this matter for the construction proof?

Think about degrees of freedom before the next slide.

Why the Argument Fails for Quadrilaterals

Consider two quadrilaterals with all sides equal to 4:

• A square: all angles 90°
• A rhombus: angles 70°, 110°, 70°, 110°

Same four side lengths — but different shapes and not congruent.

Quadrilaterals can flex — four side lengths don't determine the shape.

M4: The reverse direction relies on triangle rigidity — not obvious.

Check-In: When Is Step Three Needed?

After Steps 1–2: , , .

What determines whether is on the same side of line as ?

Also: If and are mirror images, which step handles this?

Think about handedness before the next slide.

Answer: Orientation Determines Whether Reflection Is Needed

Same orientation → after Steps 1–2; no reflection needed

Opposite orientation (mirror image) → ; reflect over line

Key: Translations and rotations preserve orientation; reflections reverse it.

At most 3 rigid motions always suffice.

CPCTC as a Proof Tool

The four-step reasoning chain:

1. Identify two triangles in the figure
2. Establish congruence (find a rigid motion — or use criteria from CO.B.8)
3. Apply CPCTC to name the specific pairs you need
4. Use those congruences to reach your conclusion

M1: Step 2 must come before Step 3 — always.

Worked Proof: Midpoints Imply Segment Congruence

Given: is the midpoint of and .

Prove:

In and :

• (midpoint of )
• (midpoint of )
• (vertical angles)

Worked Proof: Applying CPCTC to Conclude Congruence

From the previous slide: , ,

By the reverse direction:

By CPCTC: ✓

Proof structure used:

• Triangles identified → matching parts listed → congruence established → CPCTC applied

Check-In: Prove Angle Equality in a Rhombus

Given: Rhombus (all four sides equal) with diagonal .

Prove: .

Your task:

1. Identify two triangles
2. List the matching parts
3. Establish congruence
4. Apply CPCTC to reach

Write the proof before advancing.

Answer: Diagonal of Rhombus Proves Angle Equality

Diagonal creates and .

• (rhombus — all sides equal)
• (rhombus — all sides equal)
• (shared side — reflexive property)

By CPCTC: ✓

Key Takeaways and Misconception Warnings

✓ iff all 6 corresponding parts match
✓ Forward direction: rigid motion ⟹ all parts match (CO.A.4 preservation)
✓ Reverse direction: all parts match ⟹ rigid motion exists (3 steps)
✓ CPCTC: one congruence gives 6 pairs — 3 sides and 3 angles

M1: Prove congruence first, then invoke CPCTC — never reverse
M2: All six parts required here — CO.B.8 brings shortcuts
M3: Correspondence from the rigid motion, not the alphabet
M4: Reverse direction relies on triangle rigidity — not obvious
M5: CPCTC applies to both sides AND angles — all six, always

Preview: Efficient Triangle Congruence Criteria in CO.B.8

CO.B.7 established: All 6 parts match ⟺ triangles congruent

CO.B.8 will prove: Fewer parts can suffice —

• SAS: Two sides and the included angle
• ASA: Two angles and the included side
• SSS: Three sides alone

Each is a shortcut to the reverse direction of CO.B.7.