The ambiguous case.

SSA

We wish to construct triangle ABC given angle A, side AB, and side BC.

Exploration 1

1. DRAW

a. 120 degree angle with a protractor.

b. 7 cm segment

c. 6 cm segment

d. 5 cm segment

2a. Copy angle A.

Now CONSTRUCT triangle(s) ABC if you can. See diagram 2a below.

angle A = 120 degrees

AB = 6

BC = 7

2b. Copy angle A.

Now CONSTRUCT triangle(s) ABC if you can. See diagram 2b below.

angle A = 120 degrees

AB = 6

BC = 6

2c. Copy angle A.

Now CONSTRUCT triangle(s) ABC if you can. See diagram 2c below.

angle A = 120 degrees

AB = 6

BC = 5

**3. Describe what you did.**

**4. What did you discover about the number of triangles you can make for 2a, 2b, and 2c?**

**5. Write down the "rules"**

Exploration 2

1. DRAW 30 degree angle with a protractor.

2. Copy angle A

3. DRAW ray AC and ray AB (see diagram 3 below)

4. CONSTRUCT a perpendicular ( also known as the altitude or h ) from AC to ray AB to form a right angle (see diagram 6b below)

5. Measure h.

6a. Let BC < h.

Now CONSTRUCT triangle(s) ABC if you can.

6b. Choose BC so that BC = h.

Now CONSTRUCT triangle(s) ABC if you can.

6c. Choose BC so that BC > AC > h.

Now CONSTRUCT triangle(s) ABC if you can.

**7. Describe what you did.**

**8. What did you discover about the number of triangles you can make for 6a, 6b and 6c?**

**9. Write down the "rules"**

Exploration 3

1. DRAW

a. 30 degree angle with a protractor.

b. 7 cm segment

c. 6 cm segment

2. Now CONSTRUCT triangle(s) ABC if you can. See the diagram 3 below.

3. CONSTRUCT the altitude h by constructing a perpendicular from point C to line AB

4. Measure h

**5. Describe what you did.**

**6. What did you discover about the number of triangles you can make?**

**7. Write down the "rules"**

Question 1

Explain why sin C is the same in both triangles in the ambiguous case (exploration 3). This is why the Law of Sines is also ambiguous in this case.

Question 2

**What did you discover about the number of triangles you can make given SSA (two sides and an angle)? Summarize the "rules" for all cases.**

**Go HERE after 3/11/11 to see if you got the "rules" correct!**

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