Final answer:
The exact measures of angles Z1 and Z2 cannot be determined without more context. In vector scenarios, trigonometry, including the Pythagorean theorem and cosine definitions, is used to find angle measures.
Step-by-step explanation:
The question involves finding the measures of angles where the measure of Z1 is equal to the measure of Z2. Without additional information about the angles or the context in which they are used, such as in vector diagrams or parallelograms, it's not possible to provide the exact measures of the angles. However, if the angles are from vector diagrams (like those from Figures 30.52, 30.55, or 4.17), trigonometry can be used to determine the angle measures. In such cases, the Pythagorean theorem or the definition of cosine for a right-angled triangle could apply. If the vectors form a right triangle with one angle being Z1 and Z2, then the cosine of the angle can be found using the adjacent side over the hypotenuse ratio.
When determining the direction of a resultant vector, a protractor could be used to measure the angle directly, but often trigonometric relationships are more useful for precise calculations. If the scenario involves adding or subtracting vectors, angles can be measured after constructing a parallelogram, as described in Figure 2.14, and using a ruler and a protractor to measure lengths and angles, respectively.