Overview
Some of the properties of a triangle hold true even if it is an oblique triangle, when none of the angles are right angles. We can solve the sides and angles of other triangles by using trigonometric ratios, as long as we have enough information.
What Information Is Necessary?
Any triangle has 6 pieces of information, covered by the three angles A, B, and C, and the lengths of the three opposite sides of the triangle a, b, c. By convention, side a is opposite angle A, side b is opposite angle B, and side C is opposite angle B. In order to solve the triangle, at least 3 pieces of information must be known, and one of those must be the length of a side. Similarly, if we know the measure of 2 sides and the included angle, we can figure out the rest of the triangle. If we know the measure of all three angles, we cannot deduce the measure of the sides, however. The triangles may be similar, but not unique. In order to solve the triangle, there can only be one possibility.
The Law of Sines
In any triangle with angles A, B, C and opposite sides a, b, c, (sin A)/a = (sin B)/b = (sin C)/c. This is because the area of the triangle ABC can be expressed by the measurement of its sines times the length of two sides, no matter what sides and angles are chosen.
Angle- Side- Angle or Side- Angle- Angle
If we know the measure of 2 angles, we can find the measure of the third, because the degree measure of the angles of a triangle add up to be 180o. That will make it possible to use the Law of Sines. This is always true whether the side is the initial side of both of the known angles or the terminal side of one of the angles. The known information forms only one triangle.
Side-Side-Angle
If two sides are known and one angle, there are three possibilities. Sometimes there is only one triangle with the information. There can be two possibilities that fit the data. Sometimes, the data presents an impossible situation. No triangle can be constructed to fit the situation.
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