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Drawing Geometric Shapes | Lesson 2 of 2

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
Grade 7 Math | 7.G.A.2
Drawing Geometric Shapes | Lesson 2 of 2

What You Will Learn Today

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

  1. Construct a triangle given two sides and the included angle (SAS) and recognize that SAS determines a unique triangle
  2. Construct a triangle given two angles and the included side (ASA) and recognize that ASA determines a unique triangle
  3. Explain why three angle measures alone (AAA) do not determine a unique triangle
  4. Analyze SSA conditions and determine whether zero, one, or two triangles can be formed
Grade 7 Math | 7.G.A.2
Drawing Geometric Shapes | Lesson 2 of 2

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
Grade 7 Math | 7.G.A.2
Drawing Geometric Shapes | Lesson 2 of 2

SAS Construction: Two Sides, Included Angle

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

SAS construction: segment AB = 6 cm, 50-degree angle drawn at A, ray with point C at 4 cm from A, BC drawn to complete triangle

  • Draw cm. At , draw a ray at . Mark at cm on that ray. Draw .
Grade 7 Math | 7.G.A.2
Drawing Geometric Shapes | Lesson 2 of 2

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
Grade 7 Math | 7.G.A.2
Drawing Geometric Shapes | Lesson 2 of 2

ASA Construction: Two Angles, Included Side

Given: , side cm,

  • Draw cm. At , ray at . At , ray at .
  • Label intersection . Triangle done.
  • Third angle:
Grade 7 Math | 7.G.A.2
Drawing Geometric Shapes | Lesson 2 of 2

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
Grade 7 Math | 7.G.A.2
Drawing Geometric Shapes | Lesson 2 of 2

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?

Grade 7 Math | 7.G.A.2
Drawing Geometric Shapes | Lesson 2 of 2

AAA Surprise: Same Angles, Different Sizes

Two triangles with identical angle measures 40-60-80 degrees but different sizes side by side — one small, one large

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

Grade 7 Math | 7.G.A.2
Drawing Geometric Shapes | Lesson 2 of 2

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.

Grade 7 Math | 7.G.A.2
Drawing Geometric Shapes | Lesson 2 of 2

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.

Grade 7 Math | 7.G.A.2
Drawing Geometric Shapes | Lesson 2 of 2

Summary Table: SSS, SAS, ASA, AAA

Condition Result Why
SSS (3 sides) Unique triangle Side lengths fix all three vertices
SAS (2 sides + included angle) Unique triangle Included angle locks the angle between sides
ASA (2 angles + included side) Unique triangle Rays from both ends intersect at only one point
AAA (3 angles) Infinitely many (similar) Angles fix shape but not size

Reference table visual: four condition diagrams side by side with unique/multiple labels

Grade 7 Math | 7.G.A.2
Drawing Geometric Shapes | Lesson 2 of 2

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
Grade 7 Math | 7.G.A.2
Drawing Geometric Shapes | Lesson 2 of 2

SSA Case 0: No Triangle Possible

SSA diagram: angle A = 30 degrees, side b = 8 cm drawn, arc at far end with radius 3 cm that falls short of the far ray

Given: , cm, cm

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

Grade 7 Math | 7.G.A.2
Drawing Geometric Shapes | Lesson 2 of 2

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: )
Grade 7 Math | 7.G.A.2
Drawing Geometric Shapes | Lesson 2 of 2

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.

Grade 7 Math | 7.G.A.2
Drawing Geometric Shapes | Lesson 2 of 2

SSA Summary: How Many Triangles?

Case Condition Triangles
Too short height from to far ray 0
Just right height (right angle at ) 1
Ambiguous height 2
Long enough 1

The height = — but at Grade 7, compass construction reveals the cases directly.

Grade 7 Math | 7.G.A.2
Drawing Geometric Shapes | Lesson 2 of 2

SSA Is Not a Congruence Criterion

SAS SSA
Angle position Between the sides Not between the sides
Result Always unique 0, 1, or 2 triangles

SSA is not a congruence theorem.

Grade 7 Math | 7.G.A.2
Drawing Geometric Shapes | Lesson 2 of 2

Practice: How Many SSA Triangles?

Determine 0, 1, or 2 triangles:

  1. , ,
  2. , ,
  3. , ,
  4. , ,
Grade 7 Math | 7.G.A.2
Drawing Geometric Shapes | Lesson 2 of 2

Answers: Check Your SSA Practice

Set Analysis Triangles
, , → arc misses 0
, , → only one valid intersection 1
, , Obtuse : only 0 or 1 possible; arc misses 0
, , height → tangent case, right angle 1
Grade 7 Math | 7.G.A.2
Drawing Geometric Shapes | Lesson 2 of 2

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

Grade 7 Math | 7.G.A.2
Drawing Geometric Shapes | Lesson 2 of 2

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.

Grade 7 Math | 7.G.A.2