To solve the problem, we are asked to prove that triangles $\triangle AMC$ and $\triangle AMB$ are congruent.

Here is the approach to prove congruence:

Step 1: Identify the Given Information

• The triangle $ABC$ is provided, and we can see point $M$ as the midpoint of side $BC$.
• From the diagram:
  • $\overline{BM} = \overline{MC}$ (since $M$ is the midpoint),
  • $\overline{AM}$ is common to both triangles $\triangle AMC$ and $\triangle AMB$,
  • Angles $\alpha$ and $\beta$ are marked as equal, which means $\angle BAM = \angle CAM$.

Step 2: Use a Congruence Criterion

We can apply the SAS (Side-Angle-Side) Congruence Theorem to prove the triangles are congruent. According to the SAS theorem, if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.

For $\triangle AMC$ and $\triangle AMB$:

• Side $\overline{AM}$ is shared between the two triangles.
• Side $\overline{BM} = \overline{MC}$ (since $M$ is the midpoint of $BC$).
• Angle $\alpha = \beta$.

Since two sides and the included angle are congruent, we conclude that: $\triangle AMC \cong \triangle AMB$

Conclusion:

By the SAS Congruence Theorem, $\triangle AMC$ is congruent to $\triangle AMB$.

Do you want more details or have any questions?

Here are 5 related questions for further practice:

1. How does the midpoint theorem assist in proving triangle congruence?
2. Can you identify another pair of triangles in the same figure that could potentially be congruent?
3. How would you prove congruence using the SSS (Side-Side-Side) theorem?
4. What changes in the problem if the angles $\alpha$ and $\beta$ were not equal?
5. Can you think of a real-world application where congruent triangles are important?

Tip: Always look for shared sides and equal angles when identifying congruent triangles!