g10-09-03 · Syllabus supplement
The same SSA data can make two triangles
How can two sides and a nonincluded angle give0,1 or2 solutions?
1 MIN · PREDICT ON PAPER
Fix A=30°,b=10. Predict the number of triangles at a=4,5,6,10.
Discuss or write first. No typed answer is required to continue.
0 nondegenerate triangle(s); each candidate needs A+B<180°.
2 MIN · EXPLAIN AND REVISE
Which condition does your claim need?
Compare at least two cases. Write a reason, then check the explanation together.
Reveal explanation and conditions
The sine rule fixes sinB=b sinA/a. Both B and180°−B may satisfy it, but each must leave C=180°−A−B positive.
Conditions: All sides and angles must be positive. A sine value above1 means no triangle; a zero remaining angle is excluded.
Mathematical justification: For0<sinB<1 there are two candidates in(0°,180°). Testing A+B<180° proves which candidates are actual triangles.
1 MIN · EXIT CHECK
With A=30°,b=10, how many triangles exist at a=6?
Check after answering
Two: both candidate B values leave a positive C.
Teacher guidance and extension
Before: Triangle angles, sine and cosine.
Ask: Which second angle has the same sine, and is there room for C?
Watch for: Keeping only the calculator principal arcsine value, or counting a degenerate second triangle.
Challenge: Explain the threshold a=b sinA geometrically as a tangent case.
Suggested timing: predict 1 min, compare 3 min, explain 2 min, exit check 1 min. Observations can suggest a claim; they do not prove it for every input.
Optional local notebook
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