https://file.aiquickdraw.com/mathbot/file/1728943182001sh9l9ol8.jpg

Let's work through the problems presented in the image.

### Problem 1:
We are given:
- A line \( m \) with point \( P \) on the line \( m \).
- \( PQ \cong PR \) (equal segments), and
- \( QS \cong RS \) (equal segments).

The goal is to prove that \( \overrightarrow{SP} \perp m \) (i.e., that the line from \( S \) to \( P \) is perpendicular to \( m \)).

#### Solution outline:
Since \( PQ = PR \) and \( QS = RS \), we are dealing with symmetry around the point \( P \). \( S \) is equidistant from both points \( Q \) and \( R \), and the condition \( PQ = PR \) suggests that \( P \) is the midpoint of line \( QR \). Therefore, line \( SP \) forms a perpendicular bisector of line \( QR \). Hence, \( SP \) is perpendicular to line \( m \).

### Problem 2:
We are given:
- \( AB \cong AC \cong BC \) and \( AD \cong BD \).
  
We need to prove that \( \overline{AM} \cong \overline{MB} \) and that \( \overline{CD} \perp \overline{AB} \).

#### Solution outline:
Since \( AB = AC = BC \), triangle \( ABC \) is equilateral. In an equilateral triangle, all medians are also altitudes and angle bisectors, so \( AM \), being a median, divides \( AB \) into two equal halves, meaning \( AM = MB \). Additionally, \( CD \), being a perpendicular drawn from a vertex to the base, will be perpendicular to side \( AB \), hence \( CD \perp AB \).

### Problem 3:
We are asked to complete the table for the proof of \( DA > AC \), given that \( \triangle ABC \) is equilateral and \( D-B-C \).

The reasoning steps already given:
1. **Given**: The problem states that triangle \( ABC \) is equilateral.
2. **?**: The property of equilateral triangles implies that all angles are equal.
3. **The measure of an exterior angle**: This tells us that the exterior angle \( \angle D \) is greater than the interior angle \( \angle C \) of \( \triangle ABC \).
4. **Substitution**: This is where the exterior angle inequality is applied.

You would need to fill in the blanks for steps 2, 4, and 5 based on the geometrical properties and the relationships between the sides and angles in the equilateral triangle.

Would you like more detailed steps for each of these proofs or clarifications on any particular part? Here are some follow-up questions to expand on the topics:

1. How is the perpendicular bisector theorem used in geometry proofs?
2. What are the properties of equilateral triangles that help in proving medians are equal?
3. How does the exterior angle theorem apply to inequalities in triangles?
4. What are the properties of a perpendicular line in coordinate geometry?
5. How does symmetry help in simplifying geometric proofs?

**Tip**: When working with equilateral triangles, always remember that all medians, altitudes, and angle bisectors coincide, which simplifies many geometric proofs.

The image contains three geometry problems involving proving perpendicularity and triangle inequalities.

Solve geometry problems involving proving perpendicular lines and properties of equilateral triangles. Includes proofs for perpendicular bisectors, medians in equilateral triangles, and inequalities using the exterior angle theorem. Suitable for students in Grades 8-10.

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