Triangle Congruence Criteria: SSS and What Fails

Deck 2 of 2 — Completing the Investigation

In this deck:

• Proving SSS using circle intersection
• Why SSA is the "ambiguous case" (and fails)
• Why AAA proves only similarity, not congruence
• The complete criteria summary table

Learning Objectives for This Deck

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

1. Explain why SSS — three equal sides — guarantees triangle congruence using circle intersections
2. Construct a counterexample showing that SSA does not guarantee congruence (the ambiguous case)
3. Explain why AAA produces similar but not necessarily congruent triangles
4. Use SSS, SAS, ASA, and AAS as proof shortcuts; recognize SSA and AAA as invalid

Recap: SAS and ASA Proven in Deck One

Deck 1 proved:

• ✓ SAS — included angle locks direction, side locks distance → at
• ✓ ASA — two angle-aimed rays intersect at one point → at
• ✓ AAS — angle sum forces third angle → becomes ASA

Deck 2 investigates:

• SSS — does knowing all three sides force congruence?
• SSA and AAA — why do these fail?

SSS Criterion: Three Sides Determine a Triangle

SSS Criterion: If all three sides of one triangle are congruent to all three sides of another, then the triangles are congruent.

• Formally: if , , , then
• No angle information needed

SSS Proof — Steps 1 and 2

Given: , ,

Step 1: Translate so that maps to .

Step 2: Rotate about so that aligns with .

• Since , the point lands on
• Now and

Same opening as SAS and ASA. Now: where is ?

SSS Key Moment — Two Circles Intersect

lies on both circles → only two options, symmetric about line .

SSS Proof Complete: Reflect if C'' Misses F

must be at one of the two circle intersection points:

• Option A: — translate + rotate suffice ✓
• Option B: = reflection of over → add one reflection
  • Reflection over fixes and , moves to ✓

In both cases, rigid motions map . ✓

Triangle Rigidity — A Physical Analogy

Why SSS works intuitively: Three side lengths completely determine a triangle's shape.

• Three rigid sticks can only form one triangle shape (or its mirror image)
• Compare with quadrilaterals: four sticks can form a square, a rhombus, or a parallelogram — no rigidity
• The two circle intersections represent the triangle and its mirror image

SSS reflects the fundamental rigidity of triangles.

Quick Check — SSS Circle Argument

In the SSS proof, we construct two circles:

• Circle 1: centered at , radius
• Circle 2: centered at , radius

Why must lie on both circles?

Why are the intersection points symmetric about line ?

Think through the geometric reasoning before advancing.

SSS Circle Answer: Both Distances Pin C-Double-Prime

• On Circle 1: → distance from matches radius ✓
• On Circle 2: → distance from matches radius ✓
• Symmetric about : line bisects the chord between the two intersection points

The two positions for are and its reflection over — both give congruent triangles.

Now the Failures: SSA Is Ambiguous

SSA: Two sides and the angle opposite one of them (non-included angle)

• Setup: Draw angle , mark side along one ray, then draw arc of radius from
• The arc may intersect the other ray of angle at two points

If there are two intersection points → two different triangles → SSA fails

Let's see this construction step by step.

SSA Ambiguous Case — Two Triangles

Both and have: side , side , angle .

They are not congruent — same SSA data, different triangles.

SSA Counterexample: Thirty Degrees, Ten, Six

Counterexample: Two non-congruent triangles with the same SSA data

• , ,
• Triangle 1: is farther along the other ray — obtuse triangle
• Triangle 2: is closer — acute triangle

Both share: side 10, side 6, and angle 30°. They are different triangles.

SSA fails because: the arc from with radius 6 intersects the other ray at two points.

Quick Check — SAS vs. SSA

Triangle P: sides 8 and 5, angle between them = 40°

Triangle Q: sides 8 and 5, angle opposite the side of 5 = 40°

• Which is SAS? Which is SSA?
• Which guarantees congruence?
• In the SSA case, how many triangles might exist?

Identify each pattern, then think about what happens geometrically.

SAS vs. SSA Answer: Position of Angle Matters

Watch out: Triangle Q might give a unique triangle in some cases — but not always. SSA is unreliable precisely because it can give 2 triangles. One failure case is enough to disqualify it.

AAA — Why Three Angles Are Not Enough

AAA: Three equal angles do not guarantee congruence.

• Two triangles with the same three angles can have different side lengths
• Same shape → similar triangles
• Same shape AND same size → congruent triangles
• AAA proves similarity, not congruence

Rigid-motion argument: Rigid motions preserve distances. If side lengths differ, no rigid motion maps one to the other.

AAA Counterexample — Two Equilateral Triangles

Both have — but they are not congruent.

AAA proves similarity (same shape). Congruence also requires same size.

AAS Valid, SSA Invalid — Contrast

Angle sum rescues AAS. No analogous property rescues SSA.

Complete Summary: All Six Criteria Evaluated

The complete answer to "which three-part combinations suffice?"

Summary: Four Valid, Two Invalid Criteria

✓ SSS: Three sides → on two circles → at most one reflection needed

✓ SAS: Included angle aims ray, side sets distance → at

✓ ASA: Two angle-aimed rays intersect → at

✓ AAS: Angle sum forces third angle → reduces to ASA

SSA fails: Arc from hits other ray at two points → two possible triangles

AAA fails: Angles say nothing about size → similar, not congruent

Next Steps — Applying the Criteria

HSG.CO.B.8 is complete. You can now:

• Explain why SAS, ASA, SSS, and AAS are valid criteria (rigid-motion proofs)
• Explain why SSA and AAA fail (specific counterexamples)
• Use these criteria as efficient shortcuts in geometric proofs

Coming next: Using congruence criteria to prove theorems about lines, angles, and triangles (HSG.CO.C.9, CO.C.10)

The criteria are now your tools — not rules to cite, but theorems you understand.