https://file.aiquickdraw.com/mathbot/file/1726938461544wnv08up5.jpg

Let's go through each of the geometry questions in the image.

### Question 11:
**Dizione is using a compass and straightedge to perform which geometric construction?**
The figure shows a construction involving a perpendicular drawn from a point on a line. 
- The answer is **B. A perpendicular bisector of a line segment**.

### Question 12:
The diagram shows points P, Q, T, and W, with a line segment parallel to another line. You are asked to determine the final step in the construction of a line parallel to \(r\) through point Q.
- The answer is **C. Draw a line through T and S**.

### Question 13:
**Given \(x \parallel y\) with transversal \(m\) and \(n\), if \(m \angle 11 = 137^\circ\), find the measure of the remaining angles.**
- This question uses properties of parallel lines and transversals. Since \(x \parallel y\), we can use alternate interior and corresponding angle theorems to find the angle measures.

Starting with:
- \(m \angle 11 = 137^\circ\), then:
  - \(m \angle 12 = 43^\circ\) (since they are supplementary angles)
  - \(m \angle 1 = 137^\circ\) (corresponding angles)
  - \(m \angle 2 = 43^\circ\) (since \(m \angle 1\) and \(m \angle 2\) are supplementary)
  - \(m \angle 13 = 137^\circ\) (since \(m \angle 11\) and \(m \angle 13\) are alternate interior angles)
  - \(m \angle 5 = 43^\circ\) (corresponding to \(m \angle 2\))
  - And similarly for the remaining angles.

### Question 14:
**Iris is constructing a line perpendicular to line \(q\). First, she places the compass on point \(K\) and draws an arc through line \(q\) that goes through two points. Iris labels the two points \(A\) and \(B\). What statement would show the next step?**
- The answer is **C. Place the compass on point A or B and draw an arc below the line**.

### Question 15:
**Given: \( \angle 2 = \angle 3 \), \( \angle 1 \) and \( \angle 2 \), \( \angle 3 \) and \( \angle 4 \) are vertical angles. Prove \( \angle 1 = \angle 4 \).**
- This is a proof problem requiring the use of congruent angles and transitive property. You drag and drop the reasons:
  1. \( \angle 2 = \angle 3 \) - **Given**.
  2. \( \angle 1 \) and \( \angle 2 \), \( \angle 3 \) and \( \angle 4 \) are vertical angles - **Given**.
  3. \( \angle 1 = \angle 2 \) and \( \angle 3 = \angle 4 \) - **Vertical Angles Theorem**.
  4. \( \angle 1 = \angle 4 \) - **Transitive Property**.

Let me know if you'd like any further details on any of the steps or explanations!

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Here are 5 related questions to expand on the topics covered:

1. What are the properties of alternate interior angles when lines are parallel?
2. How do you construct a perpendicular bisector using only a compass and straightedge?
3. How can you prove two lines are parallel using corresponding angles?
4. What is the relationship between supplementary angles on a straight line?
5. How can you use the transitive property in geometric proofs?

**Tip**: When dealing with parallel lines and a transversal, always look for corresponding, alternate interior, and supplementary angles to simplify finding unknown angles.

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Explore detailed solutions to geometry problems focusing on constructions and angle relationships. This guide includes methods for constructing parallel and perpendicular lines, understanding angle relationships with transversals, and using geometric proofs.

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