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In geometry, a triangle is a three-sided polygon with three vertices and three edges. Understanding the properties of a triangle is crucial when solving for unknown measurements within them. There are various types of triangles, such as:

• Equilateral triangles, where all sides and angles are equal.
• Isosceles triangles, with two sides and two angles equal.
• Scalene triangles, where all sides and angles are different.

In this exercise, we are given a triangle defined by two angles and one side, known as an ASA triangle (Angle-Side-Angle). The angles in a triangle always add up to 180°. Designing and labeling the triangle correctly is a key step when working with triangle geometry. It helps to visualize and plan out the steps needed to find the missing pieces, whether they are angles or sides.

Calculating angles within triangles is a foundational concept in geometry. When you're given two angles in a triangle, finding the third angle is straightforward due to the triangle angle sum property. This important property states that:

• The sum of the three angles in a triangle is always 180°.

For instance, in the exercise, given \[\angle A = 23^{\circ}, \quad \angle B = 110^{\circ}\]we calculate \[\angle C\]as \[180^{\circ} - 23^{\circ} - 110^{\circ} = 47^{\circ}\]. This ensures that you correctly complete the triangle with all angles known, forming a critical step before you can apply the Law of Sines. Angle calculation prepares the way for further determination of side lengths, ensuring all internal angles are appropriately accounted for.

The Law of Sines, an important principle in trigonometry, is used to relate the angles and sides of a triangle. This law is particularly useful in non-right-angled triangles. It states:

• The ratio of the length of a side to the sine of its opposite angle is constant for all three sides of a triangle.

The formula is expressed as:\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]In the given problem, by knowing two angles \(A = 23^{\circ}, B = 110^{\circ}\) and the side \(c = 50\), you can determine the other two sides, \(a\) and \(b\), through substitution and solving. The process involves setting up proportions and using sine values to find side lengths accurately. Trigonometric tables or calculators aid in determining values like \(\sin 23^{\circ}\) and \(\sin 47^{\circ}\), essential for these calculations. This step-by-step leverage of trigonometric ratios allows for the complete resolution of the triangle's dimensions, unlocking practical problem-solving in triangle geometry.