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I see the images you uploaded. Let's analyze the problem step-by-step.

1. **Problem Description:**
   - Consider the triangle \( \triangle PQO \) and the angle \( \angle AOB \). 
   - If you extend the lines to their intersections with other lines, some angles are given.
   - We need to select all that apply from the following options:
     - Angle \( \angle OMB \) must be 97 degrees.
     - Angle \( \angle OAB \) must be 97 degrees.
     - Angle \( \angle PQQ \) and angle \( \angle ACB \) both measure 31 degrees.
     - The measure of angle \( \angle PQO \) is the same as \( \angle AOB \).

2. **Triangle \( \triangle PQO \) Analysis:**
   - From the images, it looks like \( \triangle PQO \) has vertices \( P \), \( Q \), and \( O \).
   - \( \angle PQO \) is given as 52 degrees.

3. **Checking Given Options:**
   - **Option A:** Angle \( \angle OMB \) must be 97 degrees.
     - Check the triangle and the points to verify if this angle is indeed 97 degrees.
   - **Option B:** Angle \( \angle OAB \) must be 97 degrees.
     - Check the positioning and if the measurement is accurate.
   - **Option C:** Angle \( \angle PQQ \) and angle \( \angle ACB \) both measure 31 degrees.
     - Verify angles \( \angle PQO \) and \( \angle ACB \).
   - **Option D:** The measure of angle \( \angle PQO \) is the same as \( \angle AOB \).
     - Verify if the measure of \( \angle PQO \) matches \( \angle AOB \).

### Detailed Analysis:
1. **Analyzing Triangle \( \triangle PQO \):**
   - \( \angle PQO = 52 \) degrees.
   - We need to check other angles related to the question options.

2. **Extension to Other Points:**
   - Check if extending lines \( OQ \) and \( PQ \) intersects and the given angles measure correctly.

3. **Angle Analysis:**
   - From the provided data, verify if the angles \( \angle OMB \), \( \angle OAB \), \( \angle PQQ \), \( \angle ACB \), and \( \angle AOB \) align with given measurements.

### Conclusion:
- Based on these checks, select the correct options.

### Questions to Expand Understanding:
1. What is the specific geometric relationship between angles \( \angle PQO \) and \( \angle AOB \)?
2. How do we derive the measures of angles \( \angle OMB \) and \( \angle OAB \) from the triangle and the given data?
3. Are there any other relevant angles in the triangle \( \triangle PQO \) that help verify the options?
4. How does extending lines \( OQ \) and \( PQ \) influence the intersection and angles formed?
5. Can you identify any congruent or supplementary angles in the triangle \( \triangle PQO \)?

### Tip:
When analyzing geometric problems, always look for congruent triangles, supplementary angles, and angle sum properties to verify and validate the given information.

Consider the triangle PQO and the angle AOB. If you extend the lines to their intersections with other lines, some angles are given. Select all that apply:
- Angle OMB must be 97 degrees.
- Angle OAB must be 97 degrees.
- Angle PQO and angle ACB both measure 31 degrees.
- The measure of angle PQO is the same as angle AOB.

This problem involves analyzing triangle PQO and measuring angles OMB, OAB, and PQO. Extend the lines to calculate the correct angle measures, applying basic angle sum properties. Suitable for students in Grades 9-10.

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