To find the length of side Z (let's assume it is the side opposite to vertex Z, meaning the side with endpoints X and Y, which we'll refer to as \( \overline{XY} \)), we need to consider the information given about triangle AXYZ. Given values are:
1. \( \overline{AX} = 2.8 \) cm (this is the length of the side between vertices A and X)
2. \( m\angle ZX = 39^\circ \) (this is the measure of the angle at vertex Z, between sides Z and X)
3. \( m\angle ZY = 72^\circ \) (this is the measure of the angle at vertex Z, between sides Z and Y)
We are asked to find the length of side \( \overline{Z} \) which we've interpreted as \( \overline{XY} \).
Now, if we have a situation where triangle AXYZ is actually a triangle with vertices A, X, and Y (and Z is a typo), then it is a triangle with angles at each vertex. The sum of angles in any triangle is always 180 degrees. Given that we have two angles, we can find the third angle using this property:
\[ m\angle A + m\angle X + m\angle Y = 180^\circ \]
Substituting the given angles:
\[ m\angle A + 39^\circ + 72^\circ = 180^\circ \]
To find \( m\angle A \):
\[ m\angle A = 180^\circ - 39^\circ - 72^\circ \]
\[ m\angle A = 180^\circ - 111^\circ \]
\[ m\angle A = 69^\circ \]
So, the measure of angle A is 69 degrees.
However, with just the measure of one side (AX = 2.8 cm) and the angles, we can't directly compute the length of side \( \overline{Z} \) (interpreted as \( \overline{XY} \)) without additional information. To calculate side \( \overline{XY} \), we would need to either have:
- The length of another side and use the Law of Sines, or
- The length of another side and the included angle and use the Law of Cosines.
Since we don't have sufficient additional information, we cannot calculate the length of side \( \overline{Z} \) to the nearest tenth of a centimeter or any other measurement of precision. If additional information is provided, such as the length of another side or an included angle, we could apply trigonometric rules to find the length of the desired side.
