To solve for missing sides and angles in non-right triangles, the law of sines is a crucial tool. This method is particularly useful when dealing with specific configurations of triangles, categorized by the information provided at the start of a problem. The three main types are:

• Angle-Side-Angle (ASA): Two angles and the side between them are known.
• Side-Angle-Angle (SAA): Two angles and a side that is adjacent to them are known.
• Side-Side-Angle (SSA): Two sides and an angle that is not between them are known.

In this context, we will focus on the ASA and SAA types. To classify a triangle, sketching it based on the given angles and sides can be very helpful. For example, if you know that angle A is 30 degrees, angle C is 70 degrees, and side c is 6, you can visualize the triangle and confirm that it is an SAA triangle.

Once classified, the next step is to find the third angle using the angle sum formula, which states that the sum of the angles in a triangle equals 180 degrees. Thus, if you have angles A and C, you can find angle B as follows:

\[A + B + C = 180^\circ \implies B = 180^\circ - A - C\]

Substituting the known values gives:

\[B = 180^\circ - 30^\circ - 70^\circ = 80^\circ\]

Now that all angles are known, the next step is to find the missing sides using the law of sines, which states:

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

To find side a, you can set up the equation:

\[\frac{a}{\sin A} = \frac{c}{\sin C}\]

Substituting the known values:

\[a = \frac{c \cdot \sin A}{\sin C} = \frac{6 \cdot \sin(30^\circ)}{\sin(70^\circ)}\]

Calculating this gives:

\[a \approx 3.19\]

Next, to find side b, you can use a similar approach:

\[\frac{b}{\sin B} = \frac{c}{\sin C}\]

Substituting the known values:

\[b = \frac{c \cdot \sin B}{\sin C} = \frac{6 \cdot \sin(80^\circ)}{\sin(70^\circ)}\]

Calculating this results in:

\[b \approx 6.29\]

In summary, by sketching the triangle, applying the angle sum formula, and utilizing the law of sines, you can effectively solve for all missing sides and angles in a triangle. This systematic approach is essential for mastering triangle problems in trigonometry.