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The image contains a geometry question about a parallelogram \( \square MATH \) with labeled vertices \( M, A, T, \) and \( H \), and several statements that involve proportional relationships and congruences.

### Steps to Solve:
1. **Figure Understanding**: The given quadrilateral \( \square MATH \) is a parallelogram.
   - Opposite sides are equal and parallel (\( \overline{MA} \parallel \overline{HT}, \overline{MH} \parallel \overline{AT} \)).
   - Diagonals bisect each other at \( S \).

2. **Statements Analysis**: We are asked to verify and compute relationships based on this parallelogram's properties.

#### Explanation of Statements:
1. \( \overline{MA} \equiv \overline{HT} \): True, opposite sides of a parallelogram are congruent.
2. \( \overline{MH} \equiv \overline{AT} \): True, opposite sides are congruent.
3. \( \angle MS \equiv \angle AS \): False unless explicitly stated; diagonals bisect angles only in a rhombus.
4. \( \triangle ATH \equiv \triangle MHT \): Depends on properties. Diagonals bisect parallelograms, so parts can be congruent by SSS.
5. \( \triangle LATH \equiv \dots\Apologies for the abrupt cut-off. Here's the continuation and detailed explanation of the problem:

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### Restating the Problem and Solving:
The quadrilateral \( \square MATH \) is a parallelogram. Below, I address the statements given in the problem:

1. **Statement 1: \( \overline{MA} \equiv \overline{HT} \)**  
   - **True.** In any parallelogram, opposite sides are congruent.  
   \[ \overline{MA} = \overline{HT} \]

2. **Statement 2: \( \overline{MH} \equiv \overline{AT} \)**  
   - **True.** Similar to the above, opposite sides of \( \square MATH \) are congruent.  
   \[ \overline{MH} = \overline{AT} \]

3. **Statement 3: \( \angle MS \equiv \angle AS \)**  
   - **False in general.** In parallelograms, the diagonals bisect each other, but they do not necessarily bisect the angles. This property holds only in special cases (like rhombuses or squares).

4. **Statement 4: \( \triangle ATH \equiv \triangle MHT \)**  
   - **True.**  
     Diagonals of a parallelogram bisect each other, so \( \triangle ATH \) and \( \triangle MHT \) are congruent by **SSS (Side-Side-Side)**:  
     - \( \overline{AT} = \overline{MH} \) (opposite sides).  
     - \( \overline{AH} = \overline{HT} \) (half of diagonals as \( S \) is the midpoint).  
     - \( \overline{AS} = \overline{MS} \) (halves of diagonals).  

5. **Statement 5: \( \triangle LATH \equiv \ldots \)**  
   - The specific wording is incomplete or unclear, but typically \( \triangle LATH \) might refer to larger congruences or properties derived by bisecting angles or sides.

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### Subsequent Statements (Relationships):
For numerical questions (like 6-10), they depend on proportional reasoning. Using the parallelogram properties:
1. \( \overline{MS} = \overline{SM} = \text{half diagonal.} \)  
2. Ratios such as \( \text{if } \overline{MS} = 9, \overline{SM} = \ldots \), are straightforward.

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### Let me know:
Would you like detailed solutions for statements 6-10, or clarifications on geometry properties?

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### Five Related Questions:
1. What are the properties of diagonals in parallelograms?  
2. In what cases do diagonals bisect the angles of a parallelogram?  
3. How do we prove triangles congruent using properties of parallelograms?  
4. What special parallelograms exist, and what are their unique properties?  
5. How can proportions of diagonals be calculated?

### Tip:
Always start geometry problems by marking known properties on the diagram. This helps in identifying relationships quickly.

Given a parallelogram MATH, analyze and solve statements involving congruence and proportions based on its properties.

Explore the properties of parallelogram MATH through congruence and proportionality. Understand key geometry concepts such as opposite sides, diagonals, and triangle congruence. Suitable for students in Grades 8-10.

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