The Ambiguous Case

Two sides and the wrong angle can describe two triangles.

12 problems · about 22 minutes · G-SRT.D.11

What this lesson teaches

The student determines how many triangles satisfy a given side-side-angle configuration.

Straightforward practice. Get the method working first.

5 problems

An inverse sine gives 25°. What is the other candidate, in degrees?
Answer 155
Why 155°.
Why does the Law of Cosines avoid the ambiguity?
Answer Cosine is negative for obtuse angles, so the sign settles it.
Why Cosine changes sign at 90°, so acute and obtuse cannot be confused.
An inverse sine gives 50°. The supplement, in degrees?
Answer 130
Why 180 − 50.
A = 30°, B could be 45° or 135°. How many triangles?
Answer 2
Why Check 30 + 135 against 180.
A = 100°, B could be 40° or 140°. How many triangles?
Answer 1
Why 100 + 140 exceeds 180.

The same ideas with more to keep track of.

3 problems

The Law of Sines gives sin B = 1.4. How many triangles?
Answer 0
Why Sine never exceeds 1.
A = 40°, B = 53.4°. What is the third angle, in degrees to one decimal place?
Answer 86.6
Why 180 − 40 − 53.4.
Which configurations can ever be ambiguous?
Answer SSA
Why Only the arrangement where the angle is not between the two sides.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Put the ambiguous case check in order.
Answer 1. Use the Law of Sines to find sin B. 2. Take the inverse sine for the first candidate. 3. Subtract from 180° for the second candidate. 4. Keep the second only if the two known angles stay under 180°.
Why The supplement is found before it is tested.
sin B = 0.5 and B must be obtuse. What is B, in degrees?
Answer 150
Why The obtuse angle with sine 0.5.
The Supplement: An inverse sine gives 35°. What is the supplement, in degrees?
Answer 145 degrees
Why 145°.
The Count: A = 70° and the second candidate for B would be 130°. How many triangles are possible?
Answer 1
Why One. The supplement leaves no room for a third angle.