SAS, ASA, AAA, and the Ambiguous Case

Lesson 2 of 2

In this lesson:

• Construct triangles from SAS, ASA, and AAA conditions
• Discover why AAA does not determine a unique triangle
• Investigate SSA and the zero, one, or two triangle outcomes

What You Will Learn Today

By the end, you should be able to:

1. Construct a triangle from two sides and the included angle
2. Construct a triangle from two angles and the included side
3. Explain why AAA never gives a unique triangle
4. Determine whether SSA gives 0, 1, or 2 triangles

Recap: SSS Was Unique — What About Angles?

From Lesson 1: SSS gives a unique triangle when the triangle inequality holds.

Lesson 1 warm-up: everyone drew angles 50°, 60°, 70° — and got different sizes.

• Same angles → same shape, but any size is possible
• Adding a side length anchors the scale

SAS Construction: Two Sides, Included Angle

• Given: Side cm, included angle at , side cm

• Draw cm. At , draw a ray at . Mark at cm on that ray. Draw .

SAS Determines a Unique Triangle

Compare your SAS triangle with a neighbor's — they should be congruent.

• The included angle fixes the direction of the second side
• Marking at cm leaves no freedom for a different vertex
• Conclusion: SAS always produces exactly one triangle

ASA Construction: Two Angles, Included Side

Given: , side cm,

• Draw cm. At , ray at . At , ray at .
• Label intersection . Triangle done.
• Third angle:

ASA Also Determines a Unique Triangle

After ASA construction: all class triangles should be congruent.

• The two rays can intersect at only one point above
• Third angle always forced:
• Conclusion: ASA produces exactly one triangle

Check-In: SAS or ASA? Classify These

1. Side 8 cm, angle 45°, side 5 cm
2. Angle 30°, side 7 cm, angle 60°
3. Side 6 cm, side 4 cm, angle 55° (not between the sides)
4. Angle 70°, side 10 cm, angle 40°

Which give a unique triangle? Which are something else?

AAA Surprise: Same Angles, Different Sizes

Both triangles have angles 40°, 60°, 80°. They are similar but not congruent.

AAA: Angles Fix Shape, Not Size

Given only angles , , :

• Starting side cm → small triangle
• Starting side cm → large triangle
• Any starting side → infinitely many valid triangles

Without a side length to anchor scale, the triangle can be any size.

When AAA Has No Solution at All

Example: , , → Sum = — no triangle possible

For a valid AAA condition:

• Angles must sum to exactly
• Every angle must be strictly between and

An angle of or collapses the triangle to a line.

Summary Table: SSS, SAS, ASA, AAA

Condition Outcomes at a Glance

SSA: The Condition That Breaks the Pattern

SSA: Two sides + an angle NOT between those sides.

• Angle ; adjacent side ; opposite "swinging" side

Depending on :

• Misses the far ray → 0 triangles
• Touches once → 1 triangle
• Crosses twice → 2 triangles

SSA Case 0: No Triangle Possible

Given: , cm, cm

Side cm is too short. The arc misses the far ray of angle . No triangle forms.

SSA Case 1: Exactly One Triangle

Given: , cm, cm

• Arc from radius cm reaches the far ray at one valid point
• Second intersection contradicts (fixed angle)
• One triangle (isosceles: )

SSA Case 2: Two Triangles Possible

Given: , cm, cm

Arc from radius crosses the far ray at and .

• : acute at ✓
• : obtuse at ✓

Two valid triangles.

SSA Summary: How Many Triangles?

SSA Is Not a Congruence Criterion

SSA is not a congruence theorem.

Practice: How Many SSA Triangles?

Determine 0, 1, or 2 triangles:

1. , ,
2. , ,
3. , ,
4. , ,

Answers: Check Your SSA Practice

Key Takeaways from Lesson 2

✓ SAS + ASA: unique triangle — angles are included

✓ AAA: no side length → infinitely many similar

✓ SSA: 0, 1, or 2 triangles — check arc intersections

SSA ≠ SAS — position of the angle matters

Always look for a second arc intersection in SSA

Coming Up: Angle Relationships in Triangles

Next lessons:

• Angles on a straight line and at a point (7.G.B.5)
• Supplementary, complementary, and vertical angles
• Discovering the triangle angle sum through construction

You've been using the 180° angle sum throughout this unit — soon you'll prove it.