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When solving triangles, especially non-right triangles, the \(\(\textit{Law of Sines}\)\) is an invaluable tool. It's a mathematical rule that helps us find unknown parts of a triangle, like sides or angles, given sufficient information. The law states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant across all three sides. This can be written as:
\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]

• \(a, b,\) and \(c\) are the lengths of the sides opposite angles \(A, B,\) and \(C\).
• \(\sin A, \sin B,\) and \(\sin C\) are the sine values of these angles.

This relationship is crucial for solving triangles when you have a combination of sides and angles provided. By rearranging the formula, you can easily solve for an unknown side or angle, as demonstrated in our step-by-step solution.

Triangles have a unique and straightforward property: the sum of their internal angles is always \(180^{\circ}\). This is known as the \(\(\textit{Triangle Angle Sum Property}\)\). Whether right, acute, or obtuse, this rule applies universally.
In our given exercise, we were provided two angles, \(A = 65^{\circ}\) and \(B = 49^{\circ}\). To determine the third angle \(C\), we simply subtract the sum of \(A\) and \(B\) from \(180^{\circ}\).

• This gives \(C = 180^{\circ} - A - B\).
• Substituting the given values, \(C = 180^{\circ} - 65^{\circ} - 49^{\circ} = 66^{\circ}\).

Using this property not only ensures accuracy in triangle calculations but also helps validate other calculations by ensuring the angles always add up to \(180^{\circ}\).

Solving triangles refers to the process of calculating unknown sides and angles given a few known parts. There are several approaches, but primarily we leverage properties like the \(\(\textit{Law of Sines}\)\) and the \(\(\textit{Triangle Angle Sum}\)\). This process involves:

• \(\textbf{Identifying Known Values:}\) Start by noting all given sides and angles.
• \(\textbf{Using Angle Sum:}\) If two angles are known, use the triangle angle sum property to find the third angle.
• \(\textbf{Applying Law of Sines:}\) Use the law to find any unknown sides. Set up equations according to the format \(\frac{\text{side}}{\sin(\text{opposite angle})}\).
• \(\textbf{Calculate:}\) Use a calculator to solve these equations accurately.

These techniques, when used correctly, enable you to find all unknown elements in a triangle effectively. This systematic approach simplifies complex problems by breaking them down into manageable steps.