1. **Problem Statement:** Given parallelogram MATH with vertices M, A, T, H and diagonals intersecting at point S, find the following congruences and equalities.
2. **Recall Properties of Parallelograms:**
- Opposite sides are congruent: $MA \cong TH$ and $MT \cong AH$.
- Diagonals bisect each other, so $MS \cong ST$.
- Triangles formed by diagonals are congruent by Side-Side-Side (SSS) or Side-Angle-Side (SAS).
- Opposite angles are congruent.
3. **Step-by-step Solutions:**
1. $MA \cong TH$ because opposite sides of a parallelogram are congruent.
2. $\triangle MAH \cong \triangle T H M$ because they share side $HM$, and $MA \cong TH$, and $AH$ is common side, so by SSS, the triangles are congruent.
3. $MS \cong ST$ because diagonals bisect each other at $S$.
4. $\triangle THM \cong \triangle SAM$ because diagonals bisect and create two congruent triangles by SSS.
5. $\angle ATH \cong \angle H M A$ because opposite angles in parallelogram are congruent.
**Final answers:**
1. $MA \cong TH$
2. $\triangle MAH \cong \triangle THM$
3. $MS \cong ST$
4. $\triangle THM \cong \triangle SAM$
5. $\angle ATH \cong \angle HMA$
Parallelogram Congruences
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