Triangle Congruence Criteria: SAS and ASA

Deck 1 of 2 — Proving the Valid Criteria

In this deck:

• Why checking all six parts is inefficient
• How rigid motions prove SAS and ASA work
• Why AAS also works (it reduces to ASA)

Learning Objectives for This Deck

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

1. Explain why SAS — two sides and the included angle — is sufficient to guarantee triangle congruence using rigid motions
2. Explain why ASA — two angles and the included side — is sufficient, using rigid motions and the ray-intersection argument
3. Explain why AAS reduces to ASA via the angle sum property
4. Use SAS and ASA as efficient shortcuts in geometric proofs

From Six Parts to Fewer

You already know: two triangles are congruent all six pairs of corresponding parts match.

• This is complete, but checking six parts is slow
• In most proofs, you won't know all six parts
• Question: Which combinations of three parts are enough?

Today we answer this for SAS and ASA.

Six Three-Part Combinations to Investigate

We'll prove three work — and show two fail.

Prove It or Find a Counterexample

For each combination, we either:

1. Prove it works — show matching parts force a rigid motion to exist
2. Disprove it — construct two non-congruent triangles with the same three parts

A criterion must work in ALL cases — one failure is enough to disqualify it.

We fill in our results as we go.

SAS Criterion: Two Sides, Included Angle

SAS Criterion: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another, then the triangles are congruent.

• Included angle: the angle between the two given sides
• Formally: if , , , then

SAS Proof — Step 1: Translate

Given: , ,

Step 1: Translate so that maps to .

• Translation is a rigid motion → distances and angles preserved
• After translation: , and ,

The triangle is now positioned with one vertex at .

SAS Proof — Step 2: Rotate

Step 2: Rotate about so that ray aligns with ray .

• Since , the point lands on
• Now: and

We've matched two vertices. Where does land?

SAS Key Moment: Angle and Side Pin C''

The included angle forces to a unique location — which must be .

SAS Proof Complete: Triangle ABC Maps to DEF

Result: , so the composition of translation and rotation maps .

• No reflection was needed
• SAS conditions completely determined the position of
• Therefore: ✓

Why the included angle is essential: it "aims" ray in the correct direction. Without it, could end up anywhere on a circle — that's SSA, which fails.

Quick Check — SAS vs. SSA

In each triangle below, are the marked parts SAS or SSA?

• Triangle 1: sides of length 5 and 7, with the angle between them equal to 40°
• Triangle 2: sides of length 5 and 7, with the angle opposite the side of length 7 equal to 40°

Which guarantees congruence? Why?

Think before advancing.

SAS vs. SSA: Included Angle Is Essential

Triangle 1: SAS ✓ — angle is the included angle between the two sides

Triangle 2: SSA ✗ — angle is not between the two sides

• In SAS, the included angle locks the direction of the second side
• In SSA, the non-included angle leaves free to be in two positions
• The order of letters matters: S-A-S means the angle is between the S's

Watch out: SSA is sometimes called the "ambiguous case" — it may or may not give a unique triangle. Never use SSA to conclude congruence.

ASA Criterion: Two Angles, Included Side

ASA Criterion: If two angles and the included side of one triangle are congruent to two angles and the included side of another, then the triangles are congruent.

• Included side: the side between the two given angles
• Formally: if , , , then

ASA Proof — Steps 1 and 2

Given: , ,

Step 1: Translate so that maps to . Now .

Step 2: Rotate about so that ray aligns with ray .

• Since , the point lands on
• Now ,

Same opening as SAS — now the angle conditions are different.

ASA Key Moment — Two Rays, One Intersection

Two angles aim two rays. The rays intersect at exactly one point — which must be .

ASA Proof Complete: Both Rays Intersect at F

Result: (intersection of the two angle-aimed rays)

• , , — all three vertices matched
• No reflection needed
• Therefore: ✓

Key insight: Two angles from both endpoints of the known side aim two rays. Their intersection is unique — one point, one triangle.

Quick Check — Identify the Criterion

Two triangles share these measurements:

1. Which criterion applies — SAS or ASA?
2. What is angle ? What is angle ?
3. Why is forced?

Think through each question before advancing.

ASA Answer: Ray Intersection Forces Third Vertex

Criterion: ASA ✓

• Two angles at the endpoints of the known side
• and
• After matching and : ray from aimed at , ray from aimed at

Intersection is unique → → triangles are congruent

Note: We didn't need to know the lengths , , , or — ASA forces them to match.

Why AAS Also Works: Angle Sum Rescues It

AAS Criterion: Two angles and a non-included side

• If , , (side opposite angle , not between the two angles)

AAS reduces to ASA:

1. Two angles are given: and
2. The third angles are forced:
3. Now we have , , — this is ASA on side with angles at and

AAS vs. SSA — Why the Difference Matters

The angle sum property rescues AAS. Nothing rescues SSA.

Quick Check — AAS or Not?

Determine whether each situation gives enough information for congruence:

• Situation A: , , (side between the angles)
• Situation B: , , (side opposite )
• Situation C: , , (two sides and angle at their common vertex)

Identify each as ASA, AAS, SAS, or SSA.

Think before advancing.

ASA, AAS, SAS: All Three Situations Confirmed

Reminder: For Situation C to be SAS, angle must be between sides and . If the angle were at or instead, it would be SSA — which does not work.

Summary — SAS and ASA Proven

✓ SAS: Included angle aims ray, side length sets distance → pinned to

✓ ASA: Two angles aim two rays from endpoints of known side → rays intersect at

✓ AAS: Two angles force the third via angle sum → reduces to ASA

Watch out: The angle must be between the two sides for SAS. Non-included angle = SSA = ambiguous case (fails).

Watch out: AAS works (angle sum rescues it). SSA does not (no "side sum" property).

Next: Deck 2 — SSS and What Fails

Deck 2 covers:

• Proving SSS — three sides force the third vertex via two circle intersections
• Showing SSA fails — the non-included angle leaves two positions for
• Showing AAA fails — angles alone say nothing about size
• The complete summary table of all six criteria

Deck 2 completes the investigation and closes the criteria table.