Which set of dimensions describe an ambiguous triangle? Select one: $$ \begin{array}{l}A=340, a=8 \mathrm{~cm}, b=10 \mathrm{~cm} \ A=740, a=10 \mathrm{~cm}, b=8 \mathrm{~cm} \ A=1040, b=12 \mathrm{~cm}, c=8 \mathrm{~cm} \ a=12 \mathrm{in}, b=14 \mathrm{in}, c=20 \mathrm{in}\end{array} $$

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To determine which set of dimensions describes an ambiguous triangle, we can use the Law of Sines. An ambiguous case occurs when we have two sides and a non-included angle (SSA condition).

The Law of Sines states:

$$
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
$$

Where:
- $$ A $$ is the angle opposite side $$ a $$
- $$ B $$ is the angle opposite side $$ b $$
- $$ C $$ is the angle opposite side $$ c $$

An ambiguous case arises when:
1. The given angle $$ A $$ is acute.
2. The opposite side $$ a $$ is shorter than side $$ b $$.

Let's analyze each option:

1. **Option 1:** $$ A = 340^\circ, a = 8 \, \text{cm}, b = 10 \, \text{cm} $$
   - Here, $$ A $$ is obtuse, so it cannot be ambiguous.

2. **Option 2:** $$ A = 740^\circ, a = 10 \, \text{cm}, b = 8 \, \text{cm} $$
   - $$ A $$ is greater than $$ 180^\circ $$, which is not possible for a triangle.

3. **Option 3:** $$ A = 1040^\circ, b = 12 \, \text{cm}, c = 8 \, \text{cm} $$
   - Again, $$ A $$ is greater than $$ 180^\circ $$, which is not possible for a triangle.

4. **Option 4:** $$ a = 12 \, \text{in}, b = 14 \, \text{in}, c = 20 \, \text{in} $$
   - This does not provide an angle, but we can check if it can form a triangle using the triangle inequality:
     - $$ a + b > c $$ → $$ 12 + 14 > 20 $$ → $$ 26 > 20 $$ (True)
     - $$ a + c > b $$ → $$ 12 + 20 > 14 $$ → $$ 32 > 14 $$ (True)
     - $$ b + c > a $$ → $$ 14 + 20 > 12 $$ → $$ 34 > 12 $$ (True)
   - This set does not indicate ambiguity.

Since none of the options provide a valid ambiguous triangle condition, we can conclude that the only possible ambiguous case would be if we had a situation with two sides and an angle that could yield two different triangles.

However, based on the provided options, none of them describe an ambiguous triangle. If we had to choose the closest to an ambiguous case, it would be the one with two sides and an angle, but since all angles provided are invalid or do not meet the criteria, we cannot select any option as describing an ambiguous triangle.

Thus, the answer is that none of the provided sets describe an ambiguous triangle.
None of the provided sets describe an ambiguous triangle.

Which set of dimensions describe an ambiguous triangle? Select one: \( \begin{array}{l}A=340, a=8 \mathrm{~cm}, b=10 \mathrm{~cm} \\ A=740, a=10 \mathrm{~cm}, b=8 \mathrm{~cm} \\ A=1040, b=12 \mathrm{~cm}, c=8 \mathrm{~cm} \\ a=12 \mathrm{in}, b=14 \mathrm{in}, c=20 \mathrm{in}\end{array} \)

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