https://file.aiquickdraw.com/mathbot/file/1728937950400xavuihr4.jpg

Let's break down each of the problems based on the image.

### Problem 29:
- You are given that \(\overline{WZ}\) bisects \(\angle XWY\) and that \(\angle XWZ = 52^\circ\).
- Since \(\overline{WZ}\) is an angle bisector, this means that \(\angle ZWY = \angle XWZ\).
- Therefore, \(\angle ZWY = 52^\circ\).
- The total \(\angle XWY = \angle XWZ + \angle ZWY = 52^\circ + 52^\circ = 104^\circ\).

### Problem 30:
- Here, you are given that \(\overline{WZ}\) bisects \(\angle XWY\) and that \(\angle ZWY = 68^\circ\).
- Since \(\overline{WZ}\) bisects \(\angle XWY\), \(\angle XWZ = \angle ZWY = 68^\circ\).
- The total \(\angle XWY = \angle XWZ + \angle ZWY = 68^\circ + 68^\circ = 136^\circ\).

### Problem 31:
- In this case, you are given \(\angle XWY = 71^\circ\), and \(\overline{WZ}\) bisects \(\angle XWY\).
- Since \(\overline{WZ}\) bisects \(\angle XWY\), \(\angle XWZ = \angle ZWY\).
- Therefore, \(\angle XWZ = \angle ZWY = \frac{71^\circ}{2} = 35.5^\circ\).

### Summary of Missing Angles:
1. **Problem 29:**
   - \(\angle ZWY = 52^\circ\)
   - \(\angle XWY = 104^\circ\)

2. **Problem 30:**
   - \(\angle XWZ = 68^\circ\)
   - \(\angle XWY = 136^\circ\)

3. **Problem 31:**
   - \(\angle XWZ = 35.5^\circ\)
   - \(\angle ZWY = 35.5^\circ\)

Would you like more details on any of these problems or have any questions?

### Related Questions:
1. How do you calculate the total angle \(\angle XWY\) if given one of the bisected angles?
2. What is the relationship between an angle bisector and the two angles it creates?
3. Can you apply the angle bisector concept in a triangle to find unknown angles?
4. What is the sum of interior angles in a triangle, and how does the angle bisector affect it?
5. How would you find the missing angle in a situation where the total angle and one bisected part are known?

### Tip:
An angle bisector divides an angle into two equal smaller angles, and this property can be applied in different geometric situations, like in triangles or polygons.

Given that WZ bisects ∠XWY, find the two angle measures not given in the diagram for problems 29, 30, and 31.

Learn how to solve angle bisector problems where WZ bisects ∠XWY in three different scenarios. Detailed solutions for finding the unknown angles based on given bisected angles are provided. Suitable for middle school students, Grades 6-8.

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