Compare side-angle pairs and enumerate zero, one or two SSA triangles.
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Try
In a triangle, label angles A, B and C and their opposite sides a, b and c. Suppose A=30°, B=90° and b=12 cm. Find a using a/sin A=b/sin B. Since sin30°=1/2 and sin90°=1, side a is 6 cm. Side-angle pairing is essential: a is opposite A, not adjacent to it.
Predict
Predict whether knowing one angle and two sides always determines one triangle. If the known angle is not between the two known sides, a side may swing into two possible positions. The sine law will expose this ambiguity because supplementary angles have the same sine.
Experiment
Take A=30°, a=5 and b=8. The sine law gives sin B=b sin A/a=4/5. Let β=arcsin(4/5), approximately 53.13°. The second candidate is 180°−β, approximately 126.87°. Both leave a positive third angle after subtracting A and B from 180°, so there are two triangles.
Change a from 5 to 3. The calculated sine becomes 4/3, exceeding 1, so no triangle exists. Change a to 4: the sine becomes 1 and B=90°, giving one triangle. These changes illustrate feasibility, not numerical instability.
Understand
The sine law states a/sin A=b/sin B=c/sin C for a nondegenerate triangle. One derivation draws an altitude: the same height can be expressed using two sides and their opposite-angle sines, yielding the ratio. Triangle angles are positive and sum to 180°, or π radians; sides are positive.
For an acute given A, define h=b sin A. This is the altitude threshold. If a<h there is no triangle; if a=h there is one right triangle; if h<a<b there are two; and if a≥b there is one. At a=b, one supplementary candidate would make the third angle zero and must be rejected. These count rules assume A is acute; do not transfer them blindly to an obtuse given angle.
For an obtuse A, its opposite side a must be the longest. With A=120°, a=6 and b=4, the principal inverse-sine candidate for B is feasible, but its supplement cannot fit beside an already obtuse angle. Check the angle sum rather than automatically accepting two outputs from a sine equation.
Use radians or degrees consistently with the available angle values and calculator mode. The geometric law itself does not depend on the notation, but entering 30 as radians when the problem gives 30° changes the sine dramatically. Exact expressions such as arcsin(√3/3) can be retained rather than rounded early.
Apply
For each SSA problem, calculate the sine ratio, check whether it lies in [−1,1], generate candidate angles and reject zero or negative remaining angles. Because triangle sides are positive here, a negative sine ratio would also signal an inconsistent setup. State the number of feasible triangles before reporting missing sides.
Continue
When two sides and their included angle are known, or when all three sides are known, the cosine law supplies a more direct route. It also explains how acute and obtuse angles change the opposite side length.
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Practice and reasoning
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A triangle has A=30°, B=90° and side b=10 opposite B. Find side a.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For A=30°, a=5 and b=8, how many nondegenerate triangles exist?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For A=30°, a=3 and b=8, how many triangles exist?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Why must a sine-law calculation sometimes test a second angle?
For an acute given angle A and positive sides a,b, define h=b sin A. Explain the counts when a<h, a=h, h<a<b, and a≥b.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For A=120°, a=6 and b=4, explain why there is only one triangle even if a calculator suggests supplementary candidates. Give an exact expression for B.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
