The Ambiguous Case: When Two Sides and an Angle Don't Determine a Triangle
In a geometry class, students are asked: 'In triangles ABC and DEF, AB = DE = 9 cm, BC = EF = 5 cm, and m∠A = m∠D = 30°. Must the triangles be congruent?' Jordan writes, 'Yes - two sides and an angle match, so the triangles are congruent by SAS.' Mr. Reyes knows the item is designed to surface ambiguity, but he wants students to see why the correspondence matters, not just memorize that SAS needs the included angle. A parent has emailed asking why the answer differs from an earlier statement in class that 'two sides and an angle determine a triangle,' and Mr. Reyes has 20 minutes the next day for an investigation with compasses and rulers before students write justifications. He wants a precise sentence he can put on the board that is true and usable, and he wants to anticipate the two triangles students will draw.
1. Determine whether the triangles must be congruent. If they need not be, produce two specific non-congruent triangles meeting the conditions and verify them.
Read the worked answer
They need not be congruent. The given angle is not between the two given sides, so this is SSA, not SAS.
Use the law of sines in triangle ABC, with side a = BC = 5 opposite ∠A and side c = AB = 9 opposite ∠C:
sin C / 9 = sin 30° / 5 → sin C = 9(0.5)/5 = 0.9.
So C ≈ 64.16° or C ≈ 115.84°. Both are admissible because in each case A + C < 180°.
- Case 1: C ≈ 64.16°, B ≈ 85.84°, AC = 5 sin 85.84°/sin 30° ≈ 9.97 cm.
- Case 2: C ≈ 115.84°, B ≈ 34.16°, AC = 5 sin 34.16°/sin 30° ≈ 5.62 cm.
Both triangles have AB = 9, BC = 5, ∠A = 30°, yet their third sides differ (≈ 9.97 vs ≈ 5.62 cm), so they are not congruent. The same two options are available for triangle DEF, and nothing in the given data forces the two triangles to land on the same option.
2. Explain precisely what is wrong with Jordan's reasoning, and evaluate the earlier class statement that 'two sides and an angle determine a triangle.'
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Jordan's error is a correspondence error, not an arithmetic one. SAS names the angle included between the two sides being used. The sides given are AB and BC; the angle included between them is ∠B, not ∠A. Jordan pattern-matched the phrase 'two sides and an angle' to SAS without checking whether the named angle lies between the named sides. Because ∠A is opposite BC, the data are SSA, which is not a congruence criterion.
The earlier statement is false as worded. Two sides and a non-included angle may determine zero, one, or two triangles. With sides c (adjacent to the angle) and a (opposite it) and angle A, let h = c sin A, the perpendicular distance from the far vertex to the ray:
- a < h: no triangle;
- a = h: exactly one, a right triangle;
- h < a < c: two non-congruent triangles (the ambiguous case);
- a ≥ c: exactly one triangle.
Here h = 9 sin 30° = 4.5 and 4.5 < 5 < 9, so we are squarely in the ambiguous case. A corrected statement: 'Two sides and the included angle determine a triangle up to congruence; two sides and a non-included angle may not.'
3. Describe what students will observe in the compass construction, and write the precise board statement you would use to close the lesson.
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Construction. Draw AB = 9 cm. Construct a ray from A making 30° with AB. Set the compass to 5 cm, center it at B, and swing arcs across the ray. The perpendicular distance from B to the ray is 9 sin 30° = 4.5 cm, which is less than 5 cm, and the radius 5 cm is less than AB = 9 cm, so the circle cuts the ray in two points. Label them C₁ and C₂; measure AC₁ ≈ 9.97 cm and AC₂ ≈ 5.62 cm. Students see two genuinely different triangles from one set of measurements, and they can see the perpendicular segment of length 4.5 cm that explains why the arcs miss the ray when the radius drops below 4.5 cm.
Board statement. 'Two sides and a non-included angle do not determine a triangle: SSA can produce 0, 1, or 2 non-congruent triangles. SAS is a congruence criterion only when the angle is included between the two stated sides.'
Closing check. Ask students to sketch the SSA data for a = 4 cm with the same 9 cm side and 30° angle and state how many triangles exist (none, since 4 < 4.5). That item cannot be answered by reciting the criterion; it requires the distance comparison.