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In this proof, we are asked to prove that triangles \(\triangle ABM\) and \(\triangle CDM\) are congruent, given that:

- \(M\) is the midpoint of \(BD\),
- \(AM \cong CM\).

Here’s how the proof proceeds step-by-step:

1. **Point \(M\) is the midpoint of \(BD\)** — This is given information.
   
2. **\(\angle AMB \cong \angle CMD\)** — This is true due to the **Vertical Angles Theorem**. These two angles are vertical angles formed by the intersection of lines \(AB\) and \(CD\), and vertical angles are congruent.

3. **\(AB \cong CD\)** — Since \(M\) is the midpoint of \(BD\), this means that the segments \(BM\) and \(DM\) are congruent by the **Definition of Midpoint**.

4. **\(\triangle ABM \cong \triangle CDM\)** — Finally, the triangles are congruent by **SAS (Side-Angle-Side) Congruence Theorem**, since:
   - \(AB \cong CD\),
   - \(\angle AMB \cong \angle CMD\) (vertical angles),
   - \(BM \cong DM\) (from the midpoint property).

Here is the completed proof table:

| Statements                                     | Reasons                          |
|------------------------------------------------|----------------------------------|
| 1. Point \(M\) is the midpoint of \(BD\); \(AM \cong CM\) | 1. Given                           |
| 2. \(\angle AMB \cong \angle CMD\)              | 2. Vertical Angles Theorem        |
| 3. \(BM \cong DM\)                              | 3. Definition of midpoint         |
| 4. \(\triangle ABM \cong \triangle CDM\)        | 4. SAS Congruence Postulate       |

### Multiple Choice Explanation:
- The correct multiple-choice answers are **b. \(BM \cong DM\)** (from the definition of the midpoint) and **d. Vertical Angles Theorem** (for the congruent angles \(\angle AMB\) and \(\angle CMD\)).

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Would you like further details or have any questions?

Here are five related questions you can explore:
1. How is the SAS Congruence Theorem applied in triangle congruence proofs?
2. What other situations allow the use of vertical angles in geometric proofs?
3. How can the definition of midpoint be used to prove triangle congruence?
4. How does the Angle-Side-Angle (ASA) Congruence Theorem differ from SAS?
5. Why is the Vertical Angles Theorem essential in congruence proofs like this one?

**Tip**: Always check if vertical angles can be used when two lines intersect, as they are often a key part of congruence proofs.

Given: Point M is the midpoint of BD; AM ≅ CM. Prove: ΔABM ≅ ΔCDM.

Learn how to prove the congruence of triangles ABM and CDM using the SAS Congruence Theorem. This step-by-step proof covers the midpoint definition, vertical angles, and key congruence theorems suitable for students in Grades 9-10.

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