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Answer:
To construct triangle XYZ with the given measurements \( XY = 5 \, \text{cm} \), \( \angle XYZ = 120^\circ \), and \( YZ = 7 \, \text{cm} \), follow these steps:
1. **Draw the base \( XY \)**: Use a ruler to draw a line segment \( XY \) that is 5 cm long.
2. **Construct the \( 120^\circ \) angle at point Y**:
- Place the protractor at point Y.
- Measure and mark a \( 120^\circ \) angle from the line \( XY \).
- Draw a ray starting from point Y along this marked angle.
3. **Locate point Z**:
- Use a compass set to a radius of 7 cm (the length of \( YZ \)).
- Place the compass point at Y and draw an arc that intersects the ray drawn in step 2.
- Label the intersection point as Z.
4. **Complete the triangle**:
- Draw line segment \( XZ \) to complete triangle XYZ.
### Measuring \( \angle XYZ \) and \( \angle YXZ \)
To find \( \angle XZY \) and \( \angle YXZ \), use the fact that the sum of the angles in any triangle is \( 180^\circ \).
Given:
- \( \angle XYZ = 120^\circ \)
The remaining angles \( \angle XZY \) and \( \angle YXZ \) sum up to \( 60^\circ \) because \( 180^\circ - 120^\circ = 60^\circ \).
Next, we use the Law of Sines to find the other angles:
\[ \frac{\sin(\angle XZY)}{XY} = \frac{\sin(\angle YXZ)}{YZ} \]
Using the known lengths:
\[ \frac{\sin(\angle XZY)}{5} = \frac{\sin(120^\circ)}{7} \]
Solving for \( \sin(\angle XZY) \):
\[ \sin(\angle XZY) = \frac{5 \cdot \sin(120^\circ)}{7} \]
\[ \sin(\angle XZY) = \frac{5 \cdot \frac{\sqrt{3}}{2}}{7} \]
\[ \sin(\angle XZY) = \frac{5\sqrt{3}}{14} \]
Then, \( \angle XZY \) can be found using the inverse sine function:
\[ \angle XZY = \sin^{-1}\left(\frac{5\sqrt{3}}{14}\right) \]
Next, calculate \( \angle YXZ \) knowing that:
\[ \angle YXZ = 60^\circ - \angle XZY \]
By performing these calculations (which typically require a calculator for the inverse sine function and subtraction), we obtain the precise angle measures for \( \angle XZY \) and \( \angle YXZ \).